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Calculus 2 - Chapter 8 Cheat Sheet
Calculus 2 - Chapter 8 Cheat Sheet
Calculus 2: Chapter 8 Essentials
Sequences, Series, and Power Series
Convergence & Divergence Tests
n-th Term Test for Divergence
If $\lim_{n\to\infty} a_n \neq 0$ or DNE, then $\sum a_n$ diverges.
Pitfall:
If $\lim_{n\to\infty} a_n = 0$, the test is inconclusive. You MUST use another test.
Geometric Series
Form: $\sum_{n=0}^{\infty} ar^n$
$$ \text{Sum} = \frac{a}{1-r} $$
Converges if $|r| < 1$.
Diverges if $|r| \geq 1$.
p-Series
Form: $\sum_{n=1}^{\infty} \frac{1}{n^p}$
Converges if $p > 1$.
Diverges if $p \leq 1$.
(Case $p=1$ is the Harmonic Series).
Ratio Test
Excellent for factorials and $n$-th powers. Compute:
$$ L = \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| $$
If $L < 1$, the series converges absolutely.
If $L > 1$ or $L=\infty$, the series diverges.
If $L = 1$, the test is inconclusive.
Alternating Series Test
For $\sum (-1)^n b_n$ with $b_n > 0$.
Converges if both are true:
$b_{n+1} \le b_n$ (decreasing).
$\lim_{n\to\infty} b_n = 0$.
Power Series
Radius & Interval of Convergence
For a series $\sum c_n (x-a)^n$ centered at $a$.
Procedure:
Use the Ratio Test on the series to find an inequality involving $|x-a|$.
Solve for $|x-a| < R$.…
Calculus 2 - Chapter 8 Cheat Sheet
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<h1 class="text-4xl md:text-5xl font-bold text-gray-50">Calculus 2: Chapter 8 Essentials</h1>
<p class="text-lg text-gray-400 mt-2">Sequences, Series, and Power Series</p>
</header>
<!-- Convergence & Divergence Tests Section -->
<h2 class="section-title">Convergence & Divergence Tests</h2>
<div class="grid grid-cols-1 md:grid-cols-2 lg:grid-cols-3 gap-6">
<div class="widget">
<h3 class="widget-title">n-th Term Test for Divergence</h3>
<p>If $\lim_{n\to\infty} a_n \neq 0$ or DNE, then $\sum a_n$ <strong class="text-gray-100">diverges</strong>.</p>
<div class="pitfall">
<p class="font-semibold text-gray-200">Pitfall:</p>
<p>If $\lim_{n\to\infty} a_n = 0$, the test is inconclusive. You MUST use another test.</p>
</div>
</div>
<div class="widget">
<h3 class="widget-title">Geometric Series</h3>
<p>Form: $\sum_{n=0}^{\infty} ar^n$</p>
<div class="formula">$$ \text{Sum} = \frac{a}{1-r} $$</div>
<ul class="list-disc list-inside pl-2 space-y-1">
<li>Converges if $|r| < 1$.</li>
<li>Diverges if $|r| \geq 1$.</li>
</ul>
</div>
<div class="widget">
<h3 class="widget-title">p-Series</h3>
<p>Form: $\sum_{n=1}^{\infty} \frac{1}{n^p}$</p>
<ul class="list-disc list-inside pl-2 space-y-1 mt-4">
<li>Converges if $p > 1$.</li>
<li>Diverges if $p \leq 1$.</li>
</ul>
<p class="text-sm text-gray-400 mt-2">(Case $p=1$ is the Harmonic Series).</p>
</div>
<div class="widget lg:col-span-2">
<h3 class="widget-title">Ratio Test</h3>
<p>Excellent for factorials and $n$-th powers. Compute:</p>
<div class="formula">$$ L = \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| $$</div>
<ul class="list-disc list-inside pl-2 space-y-1">
<li>If $L < 1$, the series converges absolutely.</li>
<li>If $L > 1$ or $L=\infty$, the series diverges.</li>
<li>If $L = 1$, the test is inconclusive.</li>
</ul>
</div>
<div class="widget">
<h3 class="widget-title">Alternating Series Test</h3>
<p>For $\sum (-1)^n b_n$ with $b_n > 0$.</p>
<p class="mt-2">Converges if both are true:</p>
<ol class="list-decimal list-inside pl-2 mt-2 space-y-1">
<li>$b_{n+1} \le b_n$ (decreasing).</li>
<li>$\lim_{n\to\infty} b_n = 0$.</li>
</ol>
</div>
</div>
<!-- Power Series Section -->
<h2 class="section-title">Power Series</h2>
<div class="grid grid-cols-1 md:grid-cols-1 lg:grid-cols-1 gap-6">
<div class="widget">
<h3 class="widget-title">Radius & Interval of Convergence</h3>
<p>For a series $\sum c_n (x-a)^n$ centered at $a$.</p>
<div class="key-point">
<h4 class="font-semibold text-gray-200">Procedure:</h4>
<ol class="list-decimal list-inside text-gray-400 space-y-2 mt-2">
<li>Use the Ratio Test on the series to find an inequality involving $|x-a|$.</li>
<li>Solve for $|x-a| < R$. This $R$ is the radius of convergence.</li>
<li>Test the endpoints $x = a-R$ and $x = a+R$ separately by plugging them into the series and using other convergence tests (like p-series or AST).</li>
<li>The Interval of Convergence (I.O.C) is the interval from $a-R$ to $a+R$, including any endpoints where the series converges.</li>
</ol>
</div>
</div>
</div>
<!-- Taylor & Maclaurin Series Section -->
<h2 class="section-title">Taylor & Maclaurin Series</h2>
<div class="grid grid-cols-1 md:grid-cols-2 gap-6">
<div class="widget">
<h3 class="widget-title">Taylor Series (centered at $a$)</h3>
<div class="formula">$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n $$</div>
<p class="text-sm text-gray-400">$ = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots $</p>
</div>
<div class="widget">
<h3 class="widget-title">Maclaurin Series (Taylor at $a=0$)</h3>
<div class="formula">$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n $$</div>
<p class="text-sm text-gray-400">$ = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \dots $</p>
</div>
</div>
<!-- Key Maclaurin Series Section -->
<h2 class="section-title">Key Maclaurin Series to Memorize</h2>
<div class="grid grid-cols-1 md:grid-cols-2 lg:grid-cols-3 gap-6">
<div class="widget">
<h3 class="widget-title">Geometric Base</h3>
<div class="formula">$$ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n $$</div>
<p class="text-center text-sm text-gray-500">I.O.C: $(-1, 1)$</p>
</div>
<div class="widget">
<h3 class="widget-title">Exponential</h3>
<div class="formula">$$ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} $$</div>
<p class="text-center text-sm text-gray-500">I.O.C: $(-\infty, \infty)$</p>
</div>
<div class="widget">
<h3 class="widget-title">Sine</h3>
<div class="formula">$$ \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} $$</div>
<p class="text-center text-sm text-gray-500">I.O.C: $(-\infty, \infty)$</p>
</div>
<div class="widget">
<h3 class="widget-title">Cosine</h3>
<div class="formula">$$ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} $$</div>
<p class="text-center text-sm text-gray-500">I.O.C: $(-\infty, \infty)$</p>
</div>
<div class="widget">
<h3 class="widget-title">Natural Log</h3>
<div class="formula">$$ \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n} $$</div>
<p class="text-center text-sm text-gray-500">I.O.C: $(-1, 1]$</p>
</div>
<div class="widget">
<h3 class="widget-title">Arctangent</h3>
<div class="formula">$$ \arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} $$</div>
<p class="text-center text-sm text-gray-500">I.O.C: $[-1, 1]$</p>
</div>
</div>
<div class="mt-6">
<div class="key-point">
<h3 class="font-semibold text-gray-200">Strategy:</h3>
<p class="text-gray-400 mt-1">Create new series from these by substituting (e.g., for $e^{2x}$, replace $x$ with $2x$), differentiating, or integrating term-by-term.</p>
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