AP Calculus BC Sequences – Arithmetic, Geometric, Sandwich Theorem & Detailed Solutions
Understanding Sequences
Before studying infinite series in AP Calculus BC, we must understand sequences at a conceptual level.
A sequence is not just a list of numbers. It is a function whose inputs are positive integers:
a_n = f(n), \quad n = 1,2,3,\dots
Unlike regular functions defined on intervals, sequences are discrete. We evaluate them only at whole-number inputs.
Why Sequences Matter
In BC, sequences help us answer one central question:
What happens as n \to \infty ?
Does the sequence:
- Settle to a number? (Converge)
- Grow without bound? (Diverge)
- Oscillate forever?
Understanding this behavior prepares us for infinite series.
Arithmetic Sequences — Linear Growth Behavior
Conceptual Understanding
An arithmetic sequence grows by adding the same amount each time.
If each term increases by 3, the pattern is predictable and linear.
Mathematically:
a_{n+1} - a_n = d
The constant d/katex] is called the <b>common difference</b>.</span> </div>
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<h3 class="elementor-heading-title elementor-size-default">Deriving the Explicit Formula</h3> </div>
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<span class="elementor-heading-title elementor-size-default">Start with: a_1</span></span> </div>
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<span class="elementor-heading-title elementor-size-default">Then:</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_2 = a_1 + d</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_3 = a_1 + 2d</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_4 = a_1 + 3d</span></span> </div>
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<span class="elementor-heading-title elementor-size-default">Notice the pattern:</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = a_1 + (n-1)d</span></span> </div>
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<h3 class="elementor-heading-title elementor-size-default">Example </h3> </div>
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<span class="elementor-heading-title elementor-size-default">Let: <span class="katex-eq" data-katex-display="false">a_1 = 4, \quad d = 5</span></span> </div>
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<span class="elementor-heading-title elementor-size-default">Then:</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = 4 + 5(n-1)</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = 5n - 1</span></span> </div>
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<span class="elementor-heading-title elementor-size-default">As <span class="katex-eq" data-katex-display="false">n \to \infty</span>:</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n \to \infty</span></span> </div>
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<h3 class="elementor-heading-title elementor-size-default">Important Conclusion</h3> </div>
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<span class="elementor-heading-title elementor-size-default">Arithmetic sequences:</span> </div>
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<svg aria-hidden="true" class="e-font-icon-svg e-fas-check" viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg"><path d="M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z"></path></svg> </span>
<span class="elementor-icon-list-text">Diverge if <span class="katex-eq" data-katex-display="false">d \neq 0</span></span>
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<svg aria-hidden="true" class="e-font-icon-svg e-fas-check" viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg"><path d="M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z"></path></svg> </span>
<span class="elementor-icon-list-text">Converge only if <span class="katex-eq" data-katex-display="false">d = 0</span></span>
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<span class="elementor-heading-title elementor-size-default">This is because linear growth does not "slow down.''</span> </div>
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<h2 class="elementor-heading-title elementor-size-default">Geometric Sequences — Exponential Behavior</h2> </div>
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<h3 class="elementor-heading-title elementor-size-default">Conceptual Understanding</h3> </div>
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<span class="elementor-heading-title elementor-size-default">Instead of adding a constant amount, geometric sequences multiply by a constant ratio.</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\dfrac{a_{n+1}}{a_n} = r</span></span> </div>
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<span class="elementor-heading-title elementor-size-default">This leads to exponential growth or decay.</span> </div>
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<h3 class="elementor-heading-title elementor-size-default">Building the Formula</h3> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_1</span>
</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_2 = a_1 r</span>
</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_3 = a_1 r^2</span>
</span> </div>
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<span class="elementor-heading-title elementor-size-default">So:
</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = a_1 r^{n-1}</span></span> </div>
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<h3 class="elementor-heading-title elementor-size-default">Convergence Analysis</h3> </div>
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<span class="elementor-icon-list-text">Everything depends on <span class="katex-eq" data-katex-display="false">|r|</span>.</span>
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<span class="elementor-icon-list-icon">
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<span class="elementor-icon-list-text">If <span class="katex-eq" data-katex-display="false">|r| < 1</span>, the powers shrink toward 0.</span>
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<span class="elementor-icon-list-icon">
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<span class="elementor-icon-list-text">If <span class="katex-eq" data-katex-display="false">|r| > 1</span>, powers grow without bound.</span>
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<span class="elementor-icon-list-icon">
<svg aria-hidden="true" class="e-font-icon-svg e-fas-check" viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg"><path d="M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z"></path></svg> </span>
<span class="elementor-icon-list-text">If <span class="katex-eq" data-katex-display="false">r = -1</span>, the sequence oscillates.</span>
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<h3 class="elementor-heading-title elementor-size-default">Example </h3> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = 8\left(\dfrac34\right)^{n-1}</span></span> </div>
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<span class="elementor-heading-title elementor-size-default">Since <span class="katex-eq" data-katex-display="false">|\dfrac34|<1</span>:</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\lim_{n\to\infty} a_n = 0</span></span> </div>
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<span class="elementor-heading-title elementor-size-default">Exponential decay leads to convergence.</span> </div>
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<h2 class="elementor-heading-title elementor-size-default">Limits of Sequences</h2> </div>
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<span class="elementor-heading-title elementor-size-default">The limit of a sequence is defined as:</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\lim_{n\to\infty} a_n = L</span></span> </div>
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<span class="elementor-heading-title elementor-size-default">meaning the terms get arbitrarily close to <span class="katex-eq" data-katex-display="false">L</span>.</span> </div>
</div>
<div class="elementor-element elementor-element-e849eef elementor-invisible elementor-widget elementor-widget-heading" data-id="e849eef" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h3 class="elementor-heading-title elementor-size-default">Example — Rational Function</h3> </div>
</div>
<div class="elementor-element elementor-element-2ecb11a elementor-invisible elementor-widget elementor-widget-heading" data-id="2ecb11a" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = \dfrac{4n^2 + 3}{2n^2 - 1}</span>
</span> </div>
</div>
<div class="elementor-element elementor-element-eb6d457 elementor-invisible elementor-widget elementor-widget-heading" data-id="eb6d457" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Divide numerator and denominator by <span class="katex-eq" data-katex-display="false">n^2</span>:
</span> </div>
</div>
<div class="elementor-element elementor-element-f58ab69 elementor-invisible elementor-widget elementor-widget-heading" data-id="f58ab69" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">= \dfrac{4 + \dfrac3{n^2}}{2 - \dfrac1{n^2}}</span>
</span> </div>
</div>
<div class="elementor-element elementor-element-77b5b72 elementor-invisible elementor-widget elementor-widget-heading" data-id="77b5b72" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">As <span class="katex-eq" data-katex-display="false">n \to \infty</span>:
</span> </div>
</div>
<div class="elementor-element elementor-element-f4aa76f elementor-invisible elementor-widget elementor-widget-heading" data-id="f4aa76f" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\lim_{n\to\infty} a_n = 2</span></span> </div>
</div>
<div class="elementor-element elementor-element-39d57aa elementor-invisible elementor-widget elementor-widget-heading" data-id="39d57aa" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Highest-degree terms dominate.</span> </div>
</div>
</div>
</div>
<div data-particle_enable="false" data-particle-mobile-disabled="false" class="elementor-element elementor-element-e2c1664 e-con-full e-flex e-con e-child" data-id="e2c1664" data-element_type="container" data-e-type="container">
<div class="elementor-element elementor-element-d239128 elementor-invisible elementor-widget elementor-widget-heading" data-id="d239128" data-element_type="widget" data-e-type="widget" id="s5" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h2 class="elementor-heading-title elementor-size-default">The Sandwich (Squeeze) Theorem</h2> </div>
</div>
<div data-particle_enable="false" data-particle-mobile-disabled="false" class="elementor-element elementor-element-fc59bdd e-con-full e-flex e-con e-child" data-id="fc59bdd" data-element_type="container" data-e-type="container">
<div class="elementor-element elementor-element-15cda81 elementor-invisible elementor-widget elementor-widget-heading" data-id="15cda81" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Sometimes we cannot compute the limit directly.</span> </div>
</div>
<div class="elementor-element elementor-element-78b7b52 elementor-invisible elementor-widget elementor-widget-heading" data-id="78b7b52" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h3 class="elementor-heading-title elementor-size-default">The Idea</h3> </div>
</div>
<div class="elementor-element elementor-element-3717be0 elementor-invisible elementor-widget elementor-widget-heading" data-id="3717be0" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">If we trap a sequence between two simpler sequences that both approach the same limit, then the middle sequence must approach that limit too.</span> </div>
</div>
<div class="elementor-element elementor-element-096671d elementor-invisible elementor-widget elementor-widget-heading" data-id="096671d" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">If: <span class="katex-eq" data-katex-display="false">b_n \le a_n \le c_n</span> and <span class="katex-eq" data-katex-display="false">\lim b_n = \lim c_n = L</span>, then: <span class="katex-eq" data-katex-display="false">\lim a_n = L</span>.</span> </div>
</div>
<div class="elementor-element elementor-element-c08f5b7 elementor-invisible elementor-widget elementor-widget-heading" data-id="c08f5b7" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h3 class="elementor-heading-title elementor-size-default">Example</h3> </div>
</div>
<div class="elementor-element elementor-element-1d9e8ed elementor-invisible elementor-widget elementor-widget-heading" data-id="1d9e8ed" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = \dfrac{\sin n}{n}</span>
</span> </div>
</div>
<div class="elementor-element elementor-element-ce364a9 elementor-invisible elementor-widget elementor-widget-heading" data-id="ce364a9" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Since:</span> </div>
</div>
<div class="elementor-element elementor-element-6a63598 elementor-invisible elementor-widget elementor-widget-heading" data-id="6a63598" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">-1 \le \sin n \le 1</span>
</span> </div>
</div>
<div class="elementor-element elementor-element-f15364a elementor-invisible elementor-widget elementor-widget-heading" data-id="f15364a" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Divide by <span class="katex-eq" data-katex-display="false">n</span>:
</span> </div>
</div>
<div class="elementor-element elementor-element-4097999 elementor-invisible elementor-widget elementor-widget-heading" data-id="4097999" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">-\dfrac1n \le \frac{\sin n}{n} \le \dfrac1n</span></span> </div>
</div>
<div class="elementor-element elementor-element-ea7a8ec elementor-invisible elementor-widget elementor-widget-heading" data-id="ea7a8ec" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Both outer sequences approach 0.</span> </div>
</div>
<div class="elementor-element elementor-element-b423ed3 elementor-invisible elementor-widget elementor-widget-heading" data-id="b423ed3" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Therefore:</span> </div>
</div>
<div class="elementor-element elementor-element-786e625 elementor-invisible elementor-widget elementor-widget-heading" data-id="786e625" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\lim_{n\to\infty} \dfrac{\sin n}{n} = 0</span></span> </div>
</div>
</div>
</div>
<div data-particle_enable="false" data-particle-mobile-disabled="false" class="elementor-element elementor-element-eba6324 e-con-full e-flex e-con e-child" data-id="eba6324" data-element_type="container" data-e-type="container">
<div class="elementor-element elementor-element-7ec814e elementor-invisible elementor-widget elementor-widget-heading" data-id="7ec814e" data-element_type="widget" data-e-type="widget" id="s6" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h2 class="elementor-heading-title elementor-size-default">Absolute Value Theorem</h2> </div>
</div>
<div data-particle_enable="false" data-particle-mobile-disabled="false" class="elementor-element elementor-element-34c4f0c e-con-full e-flex e-con e-child" data-id="34c4f0c" data-element_type="container" data-e-type="container">
<div class="elementor-element elementor-element-19d09f4 elementor-invisible elementor-widget elementor-widget-heading" data-id="19d09f4" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Oscillating sequences can still converge.</span> </div>
</div>
<div class="elementor-element elementor-element-c8d5579 elementor-invisible elementor-widget elementor-widget-heading" data-id="c8d5579" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h3 class="elementor-heading-title elementor-size-default">The Principle</h3> </div>
</div>
<div class="elementor-element elementor-element-6c91538 elementor-invisible elementor-widget elementor-widget-heading" data-id="6c91538" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">If:</span> </div>
</div>
<div class="elementor-element elementor-element-e427f68 elementor-invisible elementor-widget elementor-widget-heading" data-id="e427f68" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\lim_{n\to\infty} |a_n| = 0</span></span> </div>
</div>
<div class="elementor-element elementor-element-c7cde92 elementor-invisible elementor-widget elementor-widget-heading" data-id="c7cde92" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">then:</span> </div>
</div>
<div class="elementor-element elementor-element-b85c8ab elementor-invisible elementor-widget elementor-widget-heading" data-id="b85c8ab" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\lim_{n\to\infty} a_n = 0.</span></span> </div>
</div>
<div class="elementor-element elementor-element-2cb4276 elementor-invisible elementor-widget elementor-widget-heading" data-id="2cb4276" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h3 class="elementor-heading-title elementor-size-default">Example</h3> </div>
</div>
<div class="elementor-element elementor-element-6bf67ba elementor-invisible elementor-widget elementor-widget-heading" data-id="6bf67ba" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = \dfrac{(-1)^n}{n}</span>
</span> </div>
</div>
<div class="elementor-element elementor-element-e180fea elementor-invisible elementor-widget elementor-widget-heading" data-id="e180fea" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">The signs alternate.</span> </div>
</div>
<div class="elementor-element elementor-element-f49f110 elementor-invisible elementor-widget elementor-widget-heading" data-id="f49f110" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">But:
</span> </div>
</div>
<div class="elementor-element elementor-element-c4902d7 elementor-invisible elementor-widget elementor-widget-heading" data-id="c4902d7" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">|a_n| = \dfrac1n \to 0</span>
</span> </div>
</div>
<div class="elementor-element elementor-element-8cb7cd7 elementor-invisible elementor-widget elementor-widget-heading" data-id="8cb7cd7" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Therefore:</span> </div>
</div>
<div class="elementor-element elementor-element-6ab9047 elementor-invisible elementor-widget elementor-widget-heading" data-id="6ab9047" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n \to 0</span>
</span> </div>
</div>
<div class="elementor-element elementor-element-3218fca elementor-invisible elementor-widget elementor-widget-heading" data-id="3218fca" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Oscillation does not prevent convergence if magnitude shrinks to 0.</span> </div>
</div>
</div>
</div>
<div data-particle_enable="false" data-particle-mobile-disabled="false" class="elementor-element elementor-element-5f7b555 e-con-full e-flex e-con e-child" data-id="5f7b555" data-element_type="container" data-e-type="container">
<div class="elementor-element elementor-element-c918fae elementor-invisible elementor-widget elementor-widget-heading" data-id="c918fae" data-element_type="widget" data-e-type="widget" id="s7" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h2 class="elementor-heading-title elementor-size-default">Recursive Sequence — Deep Analysis Example</h2> </div>
</div>
<div data-particle_enable="false" data-particle-mobile-disabled="false" class="elementor-element elementor-element-3c54f19 e-con-full e-flex e-con e-child" data-id="3c54f19" data-element_type="container" data-e-type="container">
<div class="elementor-element elementor-element-a1b1989 elementor-invisible elementor-widget elementor-widget-heading" data-id="a1b1989" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_1 = 2</span></span> </div>
</div>
<div class="elementor-element elementor-element-0e6d989 elementor-invisible elementor-widget elementor-widget-heading" data-id="0e6d989" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_{n+1} = \dfrac12(a_n + 6)</span></span> </div>
</div>
<div class="elementor-element elementor-element-02c7fd9 elementor-invisible elementor-widget elementor-widget-heading" data-id="02c7fd9" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Assume limit <span class="katex-eq" data-katex-display="false">L</span> exists:</span> </div>
</div>
<div class="elementor-element elementor-element-1192346 elementor-invisible elementor-widget elementor-widget-heading" data-id="1192346" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">L = \dfrac12(L+6)</span></span> </div>
</div>
<div class="elementor-element elementor-element-13ca9ef elementor-invisible elementor-widget elementor-widget-heading" data-id="13ca9ef" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Solve:</span> </div>
</div>
<div class="elementor-element elementor-element-1a401f6 elementor-invisible elementor-widget elementor-widget-heading" data-id="1a401f6" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">2L = L+6</span></span> </div>
</div>
<div class="elementor-element elementor-element-10f611f elementor-invisible elementor-widget elementor-widget-heading" data-id="10f611f" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">L=6</span></span> </div>
</div>
<div class="elementor-element elementor-element-11147f2 elementor-icon-list--layout-traditional elementor-list-item-link-full_width elementor-invisible elementor-widget elementor-widget-icon-list" data-id="11147f2" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="icon-list.default">
<div class="elementor-widget-container">
<ul class="elementor-icon-list-items">
<li class="elementor-icon-list-item">
<span class="elementor-icon-list-text">Then show:</span>
</li>
<li class="elementor-icon-list-item">
<span class="elementor-icon-list-icon">
<svg aria-hidden="true" class="e-font-icon-svg e-fas-check" viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg"><path d="M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z"></path></svg> </span>
<span class="elementor-icon-list-text">Sequence is increasing</span>
</li>
<li class="elementor-icon-list-item">
<span class="elementor-icon-list-icon">
<svg aria-hidden="true" class="e-font-icon-svg e-fas-check" viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg"><path d="M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z"></path></svg> </span>
<span class="elementor-icon-list-text">Sequence is bounded above by 6</span>
</li>
</ul>
</div>
</div>
<div class="elementor-element elementor-element-0c3547c elementor-invisible elementor-widget elementor-widget-heading" data-id="0c3547c" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Therefore, it converges to 6.</span> </div>
</div>
</div>
</div>
<div data-particle_enable="false" data-particle-mobile-disabled="false" class="elementor-element elementor-element-ac74496 e-con-full e-flex e-con e-child" data-id="ac74496" data-element_type="container" data-e-type="container">
<div class="elementor-element elementor-element-69b7193 elementor-invisible elementor-widget elementor-widget-heading" data-id="69b7193" data-element_type="widget" data-e-type="widget" id="s8" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h2 class="elementor-heading-title elementor-size-default">Monotonic and Bounded Sequences</h2> </div>
</div>
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<h3 class="elementor-heading-title elementor-size-default">Monotonic</h3> </div>
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<span class="elementor-heading-title elementor-size-default">A sequence is monotonic if it always increases or always decreases.</span> </div>
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<h3 class="elementor-heading-title elementor-size-default">Bounded</h3> </div>
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<span class="elementor-heading-title elementor-size-default">A sequence is bounded if it stays within fixed numbers.</span> </div>
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<h3 class="elementor-heading-title elementor-size-default">Monotone Convergence Theorem</h3> </div>
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<li class="elementor-icon-list-item">
<span class="elementor-icon-list-text">If a sequence is:</span>
</li>
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<span class="elementor-icon-list-icon">
<svg aria-hidden="true" class="e-font-icon-svg e-fas-check" viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg"><path d="M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z"></path></svg> </span>
<span class="elementor-icon-list-text">Monotonic</span>
</li>
<li class="elementor-icon-list-item">
<span class="elementor-icon-list-icon">
<svg aria-hidden="true" class="e-font-icon-svg e-fas-check" viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg"><path d="M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z"></path></svg> </span>
<span class="elementor-icon-list-text">Bounded</span>
</li>
</ul>
</div>
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<span class="elementor-heading-title elementor-size-default">then it must converge.
</span> </div>
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<span class="elementor-heading-title elementor-size-default">This is extremely important in recursive sequences.</span> </div>
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<h2 class="elementor-heading-title elementor-size-default">Worked Examples with Full Step-by-Step Solutions</h2> </div>
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<h3 class="elementor-heading-title elementor-size-default">Example 1: Arithmetic Sequence — Explicit Formula</h3> </div>
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<span class="elementor-heading-title elementor-size-default">Given: <span class="katex-eq" data-katex-display="false">a_1 = 7, \quad d = -3</span>
</span> </div>
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<span class="elementor-icon-list-text">Find:</span>
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<span class="elementor-icon-list-text">1. An explicit formula for <span class="katex-eq" data-katex-display="false">a_n</span></span>
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<li class="elementor-icon-list-item">
<span class="elementor-icon-list-text">2. The 12th term</span>
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<li class="elementor-icon-list-item">
<span class="elementor-icon-list-text">3. Whether the sequence converges</span>
</li>
</ul>
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<h3 class="elementor-heading-title elementor-size-default">Solution:</h3> </div>
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<span class="elementor-heading-title elementor-size-default"><b>Step 1: Use the explicit formula</span> </div>
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<span class="elementor-heading-title elementor-size-default">For arithmetic sequences:</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = a_1 + (n-1)d</span></span> </div>
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<span class="elementor-heading-title elementor-size-default">Substitute values:</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = 7 + (n-1)(-3)</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = 7 - 3(n-1)</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><b>Step 2: Simplify</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = 7 - 3n + 3</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = 10 - 3n</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><b>Step 3: Find <span class="katex-eq" data-katex-display="false">a_{12}</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_{12} = 10 - 3(12)</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_{12} = 10 - 36</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_{12} = -26</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><b>Step 4: Determine Convergence</span> </div>
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<span class="elementor-heading-title elementor-size-default">As <span class="katex-eq" data-katex-display="false">n \to \infty</span>:</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = 10 - 3n \to -\infty</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><b>Conclusion:</b> The sequence diverges.</span> </div>
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<h3 class="elementor-heading-title elementor-size-default">Example 2: Geometric Sequence — Convergence Analysis</h3> </div>
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<span class="elementor-heading-title elementor-size-default">Given: <span class="katex-eq" data-katex-display="false">a_1 = 9, \quad r = -\dfrac13</span></span> </div>
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<ul class="elementor-icon-list-items">
<li class="elementor-icon-list-item">
<span class="elementor-icon-list-text">Find:</span>
</li>
<li class="elementor-icon-list-item">
<span class="elementor-icon-list-text">1. Explicit formula</span>
</li>
<li class="elementor-icon-list-item">
<span class="elementor-icon-list-text">2. First four terms</span>
</li>
<li class="elementor-icon-list-item">
<span class="elementor-icon-list-text">3. Limit as <span class="katex-eq" data-katex-display="false">n \to \infty</span></span>
</li>
</ul>
</div>
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<h3 class="elementor-heading-title elementor-size-default">Solution:</h3> </div>
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<span class="elementor-heading-title elementor-size-default"><b>Step 1: Formula</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = a_1 r^{n-1}</span></span> </div>
</div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = 9\left(-\dfrac13\right)^{n-1}</span></span> </div>
</div>
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<span class="elementor-heading-title elementor-size-default"><b>Step 2: First Four Terms</span> </div>
</div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_1 = 9</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_2 = 9\left(-\dfrac13\right)</span></span> </div>
</div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_2 = -3</span></span> </div>
</div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_3 = -3\left(-\dfrac13\right)</span></span> </div>
</div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_3 = 1</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_4 = 1\left(-\dfrac13\right)</span></span> </div>
</div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_4 = -\dfrac13</span></span> </div>
</div>
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<span class="elementor-heading-title elementor-size-default"><b>Step 3: Determine Limit</span> </div>
</div>
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<span class="elementor-heading-title elementor-size-default">Since:
</span> </div>
</div>
<div class="elementor-element elementor-element-de25f41 elementor-invisible elementor-widget elementor-widget-heading" data-id="de25f41" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\left|-\dfrac13\right| < 1</span></span> </div>
</div>
<div class="elementor-element elementor-element-a5f92b8 elementor-invisible elementor-widget elementor-widget-heading" data-id="a5f92b8" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\left(-\dfrac13\right)^{n-1} \to 0</span></span> </div>
</div>
<div class="elementor-element elementor-element-145b686 elementor-invisible elementor-widget elementor-widget-heading" data-id="145b686" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Therefore:
</span> </div>
</div>
<div class="elementor-element elementor-element-71f5430 elementor-invisible elementor-widget elementor-widget-heading" data-id="71f5430" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\lim_{n\to\infty} a_n = 0</span></span> </div>
</div>
<div class="elementor-element elementor-element-da5163d elementor-invisible elementor-widget elementor-widget-heading" data-id="da5163d" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><b>Conclusion:</b> Sequence converges to 0 (despite oscillation).</span> </div>
</div>
</div>
</div>
<div data-particle_enable="false" data-particle-mobile-disabled="false" class="elementor-element elementor-element-7728d2f4 e-con-full e-flex e-con e-child" data-id="7728d2f4" data-element_type="container" data-e-type="container">
<div data-particle_enable="false" data-particle-mobile-disabled="false" class="elementor-element elementor-element-1cc3081b e-con-full e-flex e-con e-child" data-id="1cc3081b" data-element_type="container" data-e-type="container">
<div class="elementor-element elementor-element-bc4811e elementor-invisible elementor-widget elementor-widget-heading" data-id="bc4811e" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h3 class="elementor-heading-title elementor-size-default">Example 3: Rational Sequence — Limit</h3> </div>
</div>
<div class="elementor-element elementor-element-99de011 elementor-invisible elementor-widget elementor-widget-heading" data-id="99de011" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = \dfrac{5n^3 + 2n}{2n^3 - 7}</span></span> </div>
</div>
<div class="elementor-element elementor-element-e2ecc5a elementor-invisible elementor-widget elementor-widget-heading" data-id="e2ecc5a" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h3 class="elementor-heading-title elementor-size-default">Solution:</h3> </div>
</div>
<div class="elementor-element elementor-element-f24ab53 elementor-invisible elementor-widget elementor-widget-heading" data-id="f24ab53" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><b>Step 1: Identify Highest Power</span> </div>
</div>
<div class="elementor-element elementor-element-226c66f elementor-invisible elementor-widget elementor-widget-heading" data-id="226c66f" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Highest power in numerator and denominator is <span class="katex-eq" data-katex-display="false">n^3</span>.</span> </div>
</div>
<div class="elementor-element elementor-element-e596ecc elementor-invisible elementor-widget elementor-widget-heading" data-id="e596ecc" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><b>Step 2: Divide by <span class="katex-eq" data-katex-display="false">n^3</span></span> </div>
</div>
<div class="elementor-element elementor-element-32a3069 elementor-invisible elementor-widget elementor-widget-heading" data-id="32a3069" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n =
\dfrac{5 + \dfrac{2}{n^2}}
{2 - \dfrac{7}{n^3}}
</span></span> </div>
</div>
<div class="elementor-element elementor-element-a16a9fa elementor-invisible elementor-widget elementor-widget-heading" data-id="a16a9fa" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><b>Step 3: Take Limit</span> </div>
</div>
<div class="elementor-element elementor-element-0559d80 elementor-invisible elementor-widget elementor-widget-heading" data-id="0559d80" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">As <span class="katex-eq" data-katex-display="false">n \to \infty</span>:</span> </div>
</div>
<div class="elementor-element elementor-element-e332438 elementor-invisible elementor-widget elementor-widget-heading" data-id="e332438" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\dfrac{2}{n^2} \to 0</span></span> </div>
</div>
<div class="elementor-element elementor-element-a51ee1d elementor-invisible elementor-widget elementor-widget-heading" data-id="a51ee1d" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\dfrac{7}{n^3} \to 0</span></span> </div>
</div>
<div class="elementor-element elementor-element-a578b44 elementor-invisible elementor-widget elementor-widget-heading" data-id="a578b44" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">So:</span> </div>
</div>
<div class="elementor-element elementor-element-e5f5280 elementor-invisible elementor-widget elementor-widget-heading" data-id="e5f5280" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\lim_{n\to\infty} a_n = \dfrac{5}{2}</span></span> </div>
</div>
<div class="elementor-element elementor-element-0eea985 elementor-invisible elementor-widget elementor-widget-heading" data-id="0eea985" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h3 class="elementor-heading-title elementor-size-default">Example 4: Sandwich Theorem Application</h3> </div>
</div>
<div class="elementor-element elementor-element-35f3c96 elementor-invisible elementor-widget elementor-widget-heading" data-id="35f3c96" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = \dfrac{\cos(3n)}{n}</span></span> </div>
</div>
<div class="elementor-element elementor-element-c164150 elementor-invisible elementor-widget elementor-widget-heading" data-id="c164150" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h3 class="elementor-heading-title elementor-size-default">Solution:</h3> </div>
</div>
<div class="elementor-element elementor-element-bc0d3f8 elementor-invisible elementor-widget elementor-widget-heading" data-id="bc0d3f8" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><b>Step 1: Use Trigonometric Bounds</span> </div>
</div>
<div class="elementor-element elementor-element-aa35131 elementor-invisible elementor-widget elementor-widget-heading" data-id="aa35131" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">We know:</span> </div>
</div>
<div class="elementor-element elementor-element-bd794db elementor-invisible elementor-widget elementor-widget-heading" data-id="bd794db" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">-1 \le \cos(3n) \le 1</span></span> </div>
</div>
<div class="elementor-element elementor-element-9713351 elementor-invisible elementor-widget elementor-widget-heading" data-id="9713351" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><b>Step 2: Divide by <span class="katex-eq" data-katex-display="false">n[katex]</span> </div>
</div>
<div class="elementor-element elementor-element-5a753eb elementor-invisible elementor-widget elementor-widget-heading" data-id="5a753eb" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">[katex]-\dfrac1n \le \dfrac{\cos(3n)}{n} \le \dfrac1n</span></span> </div>
</div>
<div class="elementor-element elementor-element-8fc9b85 elementor-invisible elementor-widget elementor-widget-heading" data-id="8fc9b85" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><b>Step 3: Take Limits of Bounds</span> </div>
</div>
<div class="elementor-element elementor-element-af90e17 elementor-invisible elementor-widget elementor-widget-heading" data-id="af90e17" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">-\dfrac1n \to 0</span></span> </div>
</div>
<div class="elementor-element elementor-element-164eeb3 elementor-invisible elementor-widget elementor-widget-heading" data-id="164eeb3" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\dfrac1n \to 0</span></span> </div>
</div>
<div class="elementor-element elementor-element-75bbd1a elementor-invisible elementor-widget elementor-widget-heading" data-id="75bbd1a" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><b>Step 4: Apply Sandwich Theorem</span> </div>
</div>
<div class="elementor-element elementor-element-367132d elementor-invisible elementor-widget elementor-widget-heading" data-id="367132d" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\lim_{n\to\infty} \dfrac{\cos(3n)}{n} = 0</span></span> </div>
</div>
</div>
</div>
<div data-particle_enable="false" data-particle-mobile-disabled="false" class="elementor-element elementor-element-7b644a4 e-con-full e-flex e-con e-child" data-id="7b644a4" data-element_type="container" data-e-type="container">
<div data-particle_enable="false" data-particle-mobile-disabled="false" class="elementor-element elementor-element-aea04f0 e-con-full e-flex e-con e-child" data-id="aea04f0" data-element_type="container" data-e-type="container">
<div class="elementor-element elementor-element-e23e467 elementor-invisible elementor-widget elementor-widget-heading" data-id="e23e467" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h3 class="elementor-heading-title elementor-size-default">Example 5: Absolute Value Theorem</h3> </div>
</div>
<div class="elementor-element elementor-element-207627a elementor-invisible elementor-widget elementor-widget-heading" data-id="207627a" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = \dfrac{(-1)^n \sqrt{n}}{n}</span></span> </div>
</div>
<div class="elementor-element elementor-element-5135cfd elementor-invisible elementor-widget elementor-widget-heading" data-id="5135cfd" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h3 class="elementor-heading-title elementor-size-default">Solution:</h3> </div>
</div>
<div class="elementor-element elementor-element-b34e82f elementor-invisible elementor-widget elementor-widget-heading" data-id="b34e82f" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><b>Step 1: Simplify</span> </div>
</div>
<div class="elementor-element elementor-element-c279c4b elementor-invisible elementor-widget elementor-widget-heading" data-id="c279c4b" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n = \dfrac{(-1)^n}{\sqrt{n}}</span>.</span> </div>
</div>
<div class="elementor-element elementor-element-adff11c elementor-invisible elementor-widget elementor-widget-heading" data-id="adff11c" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><b>Step 2: Take Absolute Value</span> </div>
</div>
<div class="elementor-element elementor-element-da2d525 elementor-invisible elementor-widget elementor-widget-heading" data-id="da2d525" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">|a_n| = \dfrac{1}{\sqrt{n}}
</span></span> </div>
</div>
<div class="elementor-element elementor-element-1f952e1 elementor-invisible elementor-widget elementor-widget-heading" data-id="1f952e1" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><b>Step 3: Take Limit</span> </div>
</div>
<div class="elementor-element elementor-element-b6f3411 elementor-invisible elementor-widget elementor-widget-heading" data-id="b6f3411" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"> <span class="katex-eq" data-katex-display="false">\dfrac{1}{\sqrt{n}} \to 0</span></span> </div>
</div>
<div class="elementor-element elementor-element-7f5f69a elementor-invisible elementor-widget elementor-widget-heading" data-id="7f5f69a" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><b>Step 4: Apply Absolute Value Theorem</span> </div>
</div>
<div class="elementor-element elementor-element-413a1dd elementor-invisible elementor-widget elementor-widget-heading" data-id="413a1dd" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Since <span class="katex-eq" data-katex-display="false">|a_n| \to 0</span>,</span> </div>
</div>
<div class="elementor-element elementor-element-42af292 elementor-invisible elementor-widget elementor-widget-heading" data-id="42af292" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_n \to 0</span></span> </div>
</div>
<div class="elementor-element elementor-element-aaa9eac elementor-invisible elementor-widget elementor-widget-heading" data-id="aaa9eac" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h3 class="elementor-heading-title elementor-size-default">Example 6: Recursive Sequence — Full Justification</h3> </div>
</div>
<div class="elementor-element elementor-element-0a92e42 elementor-invisible elementor-widget elementor-widget-heading" data-id="0a92e42" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_1 = 4</span></span> </div>
</div>
<div class="elementor-element elementor-element-00ad39f elementor-invisible elementor-widget elementor-widget-heading" data-id="00ad39f" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_{n+1} = \dfrac12(a_n + 8)</span></span> </div>
</div>
<div class="elementor-element elementor-element-5b80b5f elementor-invisible elementor-widget elementor-widget-heading" data-id="5b80b5f" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<h3 class="elementor-heading-title elementor-size-default">Solution:</h3> </div>
</div>
<div class="elementor-element elementor-element-0d9a227 elementor-invisible elementor-widget elementor-widget-heading" data-id="0d9a227" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><b>Step 1: Assume Limit Exists</span> </div>
</div>
<div class="elementor-element elementor-element-f46ff06 elementor-invisible elementor-widget elementor-widget-heading" data-id="f46ff06" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default">Let:</span> </div>
</div>
<div class="elementor-element elementor-element-9add686 elementor-invisible elementor-widget elementor-widget-heading" data-id="9add686" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\lim a_n = L</span></span> </div>
</div>
<div class="elementor-element elementor-element-6cbce06 elementor-invisible elementor-widget elementor-widget-heading" data-id="6cbce06" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><b>Step 2: Substitute into Recursion</span> </div>
</div>
<div class="elementor-element elementor-element-1615666 elementor-invisible elementor-widget elementor-widget-heading" data-id="1615666" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">L = \dfrac12(L+8)</span></span> </div>
</div>
<div class="elementor-element elementor-element-5086e8a elementor-invisible elementor-widget elementor-widget-heading" data-id="5086e8a" data-element_type="widget" data-e-type="widget" data-settings="{"_animation":"fadeInUp"}" data-widget_type="heading.default">
<div class="elementor-widget-container">
<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">2L = L + 8</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">L = 8</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><b>Step 3: Show Boundedness</span> </div>
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<span class="elementor-heading-title elementor-size-default">Assume <span class="katex-eq" data-katex-display="false">a_n < 8</span>.</span> </div>
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<span class="elementor-heading-title elementor-size-default">Then:</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_{n+1} = \dfrac12(a_n + 8) < 8</span></span> </div>
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<span class="elementor-heading-title elementor-size-default">So sequence is bounded above by 8.</span> </div>
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<span class="elementor-heading-title elementor-size-default"><b>Step 4: Show Monotonicity</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">a_{n+1} - a_n =
\dfrac12(a_n+8) - a_n</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">= \dfrac12a_n + 4 - a_n</span></span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">= 4 - \dfrac12 a_n</span></span> </div>
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<span class="elementor-heading-title elementor-size-default">If <span class="katex-eq" data-katex-display="false">a_n < 8</span>, then:</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">4 - \dfrac12 a_n > 0</span></span> </div>
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<span class="elementor-heading-title elementor-size-default">Thus sequence is increasing.</span> </div>
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<span class="elementor-heading-title elementor-size-default"><b>Final Conclusion</span> </div>
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<ul class="elementor-icon-list-items">
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<span class="elementor-icon-list-text">Sequence is:</span>
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<span class="elementor-icon-list-icon">
<svg aria-hidden="true" class="e-font-icon-svg e-fas-check" viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg"><path d="M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z"></path></svg> </span>
<span class="elementor-icon-list-text">Increasing</span>
</li>
<li class="elementor-icon-list-item">
<span class="elementor-icon-list-icon">
<svg aria-hidden="true" class="e-font-icon-svg e-fas-check" viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg"><path d="M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z"></path></svg> </span>
<span class="elementor-icon-list-text">Bounded above by 8</span>
</li>
</ul>
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<span class="elementor-heading-title elementor-size-default">By Monotone Convergence Theorem:</span> </div>
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<span class="elementor-heading-title elementor-size-default"><span class="katex-eq" data-katex-display="false">\lim a_n = 8</span></span> </div>
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<h2 class="elementor-heading-title elementor-size-default">Frequently Asked Questions </h2> </div>
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<span class='e-n-accordion-item-title-header'><div class="e-n-accordion-item-title-text"> Q1: Why do arithmetic sequences usually diverge? </div></span>
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<span class='e-opened' ><svg aria-hidden="true" class="e-font-icon-svg e-fas-chevron-down" viewBox="0 0 448 512" xmlns="http://www.w3.org/2000/svg"><path d="M207.029 381.476L12.686 187.132c-9.373-9.373-9.373-24.569 0-33.941l22.667-22.667c9.357-9.357 24.522-9.375 33.901-.04L224 284.505l154.745-154.021c9.379-9.335 24.544-9.317 33.901.04l22.667 22.667c9.373 9.373 9.373 24.569 0 33.941L240.971 381.476c-9.373 9.372-24.569 9.372-33.942 0z"></path></svg></span>
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<span class="elementor-heading-title elementor-size-default">Because linear growth continues without slowing down.</span> </div>
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<span class='e-n-accordion-item-title-header'><div class="e-n-accordion-item-title-text"> Q2: Why does [katex]|r|<1 guarantee convergence in geometric sequences?
Because repeated multiplication shrinks the terms toward 0.
Q3: Why is the Sandwich Theorem important?
It allows us to evaluate limits that cannot be computed directly.
Q4: What role does absolute value play in convergence?
It measures size regardless of sign.
Q5: Why are sequences essential for series?
Because series are sums of sequence terms.
Final Takeaway
To master AP Calculus BC sequences, you must understand:
- Linear vs exponential growth
- Long-term limit behavior
- Bounding arguments
- Oscillation control
- Recursive convergence logic
At mathaversity we help students do well in AP Calculus BC by giving them personalized lessons, practice sessions, and tips for taking tests. Make an appointment for your free demo today and improve your AP Calculus skills!
- Quick Links:
- Why Sequences Matter
- Arithmetic Sequences — Linear Growth Behavior
- Geometric Sequences — Exponential Behavior
- Limits of Sequences
- The Sandwich (Squeeze) Theorem
- Absolute Value Theorem
- Recursive Sequence — Deep Analysis Example
- Monotonic and Bounded Sequences
- Worked Examples with Full Step-by-Step Solutions
- Frequently Asked Questions
- Final Takeaway
