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Calc 2 God: Power Series Cheat Sheet
Calc 2 God: Power Series Cheat Sheet
Power Series: The Essentials
// Your cheat sheet for the most important formulas and concepts. No fluff.
Core Concepts
General Form: A power series is centered at a.
∑ cn(x - a)n
Radius of Convergence (R): The distance from the center that the series converges.
Interval of Convergence (I.O.C.): The full range of x values where the series converges.
Most Important Rule: You must ALWAYS TEST THE ENDPOINTS of the interval.
The Ratio Test: Your Main Tool
Use the Ratio Test to find the radius. The series converges if L < 1.
L = limn→∞ |
an+1
an
|
Three Possible Outcomes:
If L = 0, the series converges for all x. Then R = ∞.
If L = ∞, the series converges only at its center, x = a. Then R = 0.
If L = K |x - a|, then the radius is R = 1/K.
Taylor & Maclaurin Series
The formula for representing a function as a power series.
Taylor Series centered at a:
f(x) = ∑
f(n)(a)
n!
(x - a)n
Maclaurin Series is a Taylor Series centered at a = 0.
Essential Maclaurin Series to Memorize
Function
Series Expansion
Interval
11 - x
1 + x + x2 + x3 + ...
(-1, 1)
ex
1 + x + x2/2! + x3/3! + ...
(-∞, ∞)
sin(x)
x - x3/3! + x5/5! - ...
(-∞, ∞)
cos(x)
1 - x2/2! + x4/4! - ...
(-∞, ∞)
ln(1 + x)
x - x2/2 + x3/3 - ...
(-1, 1]
arctan(x)
x - x3/3 + x5/5 - ...
[-1, 1]
Key Rules of Thumb ("Tricks")
The center (a) comes from the (x - a)n term.
The radius (R) comes from the other exponential parts (like kn).
Polynomial parts like nk do not affect the radius.
If you see n! in the denominator, the radius is almost always R = ∞.
If you see n! in the numerator, the radius is almost always R = 0.
Calc 2 God: Power Series Cheat Sheet
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<h1 class="text-4xl text-white mb-2" style="text-shadow: 0 0 10px rgba(88, 166, 255, 0.5);">Power Series: The Essentials</h1>
<p class="text-slate-400 mb-8">// Your cheat sheet for the most important formulas and concepts. No fluff.</p>
<!-- Core Concepts -->
<div class="mb-8">
<h2 class="text-2xl text-slate-100 border-b-2 border-blue-500 pb-2 mb-4">Core Concepts</h2>
<ul class="list-disc list-inside space-y-2 text-slate-300">
<li><b>General Form:</b> A power series is centered at <span class="symbol"><i>a</i></span>.
<div class="formula"><span class="symbol">∑</span> <i>c</i><sub>n</sub>(<i>x</i> - <i>a</i>)<sup>n</sup></div>
</li>
<li><b>Radius of Convergence (<span class="symbol"><i>R</i></span>):</b> The distance from the center that the series converges.</li>
<li><b>Interval of Convergence (I.O.C.):</b> The full range of <i>x</i> values where the series converges.</li>
<li class="key-idea"><b>Most Important Rule:</b> You must <b>ALWAYS TEST THE ENDPOINTS</b> of the interval.</li>
</ul>
</div>
<!-- The Ratio Test -->
<div class="mb-8">
<h2 class="text-2xl text-slate-100 border-b-2 border-blue-500 pb-2 mb-4">The Ratio Test: Your Main Tool</h2>
<p class="mb-2">Use the Ratio Test to find the radius. The series converges if <span class="symbol"><i>L</i> < 1</span>.</p>
<div class="formula">
<span class="symbol"><i>L</i> = lim<sub><i>n</i>→∞</sub> |</span>
<div class="fraction">
<span class="numerator"><i>a</i><sub>n+1</sub></span>
<span class="denominator"><i>a</i><sub>n</sub></span>
</div>
<span class="symbol">|</span>
</div>
<h3 class="text-xl font-semibold mt-6 mb-2 text-slate-200">Three Possible Outcomes:</h3>
<ol class="list-decimal list-inside space-y-2">
<li>If <b><span class="symbol"><i>L</i> = 0</span></b>, the series converges for all <i>x</i>. Then <b><span class="symbol"><i>R</i> = ∞</span></b>.</li>
<li>If <b><span class="symbol"><i>L</i> = ∞</span></b>, the series converges only at its center, <span class="symbol"><i>x</i> = <i>a</i></span>. Then <b><span class="symbol"><i>R</i> = 0</span></b>.</li>
<li>If <b><span class="symbol"><i>L</i> = <i>K</i> |<i>x</i> - <i>a</i>|</span></b>, then the radius is <b><span class="symbol"><i>R</i> = 1/<i>K</i></span></b>.</li>
</ol>
</div>
<!-- Taylor & Maclaurin Series -->
<div class="mb-8">
<h2 class="text-2xl text-slate-100 border-b-2 border-blue-500 pb-2 mb-4">Taylor & Maclaurin Series</h2>
<p class="mb-2">The formula for representing a function as a power series.</p>
<ul class="list-disc list-inside space-y-2">
<li><b>Taylor Series centered at <span class="symbol"><i>a</i></span>:</b>
<div class="formula">
<span class="symbol"><i>f(x)</i> = ∑</span>
<div class="fraction">
<span class="numerator"><i>f</i><sup>(n)</sup>(<i>a</i>)</span>
<span class="denominator"><i>n</i>!</span>
</div>
(<i>x</i> - <i>a</i>)<sup>n</sup>
</div>
</li>
<li><b>Maclaurin Series</b> is a Taylor Series centered at <b><span class="symbol"><i>a</i> = 0</span></b>.</li>
</ul>
</div>
<!-- Essential Maclaurin Series -->
<div class="mb-8">
<h2 class="text-2xl text-slate-100 border-b-2 border-blue-500 pb-2 mb-4">Essential Maclaurin Series to Memorize</h2>
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<tr>
<th class="p-3 rounded-tl-lg">Function</th>
<th class="p-3">Series Expansion</th>
<th class="p-3 rounded-tr-lg">Interval</th>
</tr>
</thead>
<tbody class="bg-[#161b22]">
<tr class="border-b border-slate-700">
<td class="p-3"><div class="fraction"><span class="numerator">1</span><span class="denominator">1 - <i>x</i></span></div></td>
<td class="p-3 font-mono text-green-400">1 + x + x<sup>2</sup> + x<sup>3</sup> + ...</td>
<td class="p-3 font-mono text-purple-400">(-1, 1)</td>
</tr>
<tr class="border-b border-slate-700">
<td class="p-3"><span class="symbol"><i>e</i><sup><i>x</i></sup></span></td>
<td class="p-3 font-mono text-green-400">1 + x + x<sup>2</sup>/2! + x<sup>3</sup>/3! + ...</td>
<td class="p-3 font-mono text-purple-400">(-∞, ∞)</td>
</tr>
<tr class="border-b border-slate-700">
<td class="p-3"><span class="symbol">sin(<i>x</i>)</span></td>
<td class="p-3 font-mono text-green-400">x - x<sup>3</sup>/3! + x<sup>5</sup>/5! - ...</td>
<td class="p-3 font-mono text-purple-400">(-∞, ∞)</td>
</tr>
<tr class="border-b border-slate-700">
<td class="p-3"><span class="symbol">cos(<i>x</i>)</span></td>
<td class="p-3 font-mono text-green-400">1 - x<sup>2</sup>/2! + x<sup>4</sup>/4! - ...</td>
<td class="p-3 font-mono text-purple-400">(-∞, ∞)</td>
</tr>
<tr class="border-b border-slate-700">
<td class="p-3"><span class="symbol">ln(1 + <i>x</i>)</span></td>
<td class="p-3 font-mono text-green-400">x - x<sup>2</sup>/2 + x<sup>3</sup>/3 - ...</td>
<td class="p-3 font-mono text-purple-400">(-1, 1]</td>
</tr>
<tr>
<td class="p-3 rounded-bl-lg"><span class="symbol">arctan(<i>x</i>)</span></td>
<td class="p-3 font-mono text-green-400">x - x<sup>3</sup>/3 + x<sup>5</sup>/5 - ...</td>
<td class="p-3 rounded-br-lg font-mono text-purple-400">[-1, 1]</td>
</tr>
</tbody>
</table>
</div>
</div>
<!-- Rules of Thumb -->
<div>
<h2 class="text-2xl text-slate-100 border-b-2 border-blue-500 pb-2 mb-4">Key Rules of Thumb ("Tricks")</h2>
<ul class="list-disc list-inside space-y-3">
<li>The <b>center (<span class="symbol"><i>a</i></span>)</b> comes from the <span class="font-mono text-green-400">(x - a)<sup>n</sup></span> term.</li>
<li>The <b>radius (<span class="symbol"><i>R</i></span>)</b> comes from the other exponential parts (like <span class="font-mono text-green-400">k<sup>n</sup></span>).</li>
<li>Polynomial parts like <span class="font-mono text-green-400">n<sup>k</sup></span> do not affect the radius.</li>
<li class="key-idea">If you see <b><span class="symbol"><i>n</i>!</span> in the denominator</b>, the radius is almost always <b><span class="symbol"><i>R</i> = ∞</span></b>.</li>
<li class="key-idea">If you see <b><span class="symbol"><i>n</i>!</span> in the numerator</b>, the radius is almost always <b><span class="symbol"><i>R</i> = 0</span></b>.</li>
</ul>
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