Power Series: Radius, Interval of Convergence - AP Calculus BC
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Determine the interval of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} \frac{(x-3)^n}{2^n}$.
Determine the interval of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} \frac{(x-3)^n}{2^n}$.
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$(1, 5)$. Series $\sum \frac{(x-3)^n}{2^n}$ has center 3, radius 2; endpoints diverge.
$(1, 5)$. Series $\sum \frac{(x-3)^n}{2^n}$ has center 3, radius 2; endpoints diverge.
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Determine the interval of convergence for $\textstyle\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n}$.
Determine the interval of convergence for $\textstyle\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n}$.
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$(-1, 1]$. Alternating series converges at $x = 1$; diverges at $x = -1$.
$(-1, 1]$. Alternating series converges at $x = 1$; diverges at $x = -1$.
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Determine the convergence interval of $\textstyle\sum_{n=1}^{\infty} \frac{x^n}{n}$.
Determine the convergence interval of $\textstyle\sum_{n=1}^{\infty} \frac{x^n}{n}$.
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$(-1, 1]$. Alternating harmonic-type series; converges at $x = 1$ by alternating series test.
$(-1, 1]$. Alternating harmonic-type series; converges at $x = 1$ by alternating series test.
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State the formula for the root test applied to a series.
State the formula for the root test applied to a series.
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$\textstyle\lim_{n \to \infty} \big|a_n\big|^{1/n} < 1$ for convergence. Root test condition for series convergence.
$\textstyle\lim_{n \to \infty} \big|a_n\big|^{1/n} < 1$ for convergence. Root test condition for series convergence.
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Find the interval of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} \frac{x^n}{n!}$.
Find the interval of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} \frac{x^n}{n!}$.
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$(-\text{∞}, \text{∞})$. Exponential function series; factorial dominates, giving infinite radius.
$(-\text{∞}, \text{∞})$. Exponential function series; factorial dominates, giving infinite radius.
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What is the radius of convergence if $L = 0$ in the ratio test?
What is the radius of convergence if $L = 0$ in the ratio test?
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$R = \text{∞}$. When limit is 0, radius becomes infinite.
$R = \text{∞}$. When limit is 0, radius becomes infinite.
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Identify the radius of convergence for $a_n = \frac{1}{n^2}$ using the ratio test.
Identify the radius of convergence for $a_n = \frac{1}{n^2}$ using the ratio test.
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$R = \text{∞}$. Quadratic decay gives infinite radius via ratio test.
$R = \text{∞}$. Quadratic decay gives infinite radius via ratio test.
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Which test confirms divergence if the limit is greater than 1?
Which test confirms divergence if the limit is greater than 1?
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Ratio test or root test. Both tests indicate divergence when limit exceeds 1.
Ratio test or root test. Both tests indicate divergence when limit exceeds 1.
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Identify the radius of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} 2^n x^n$.
Identify the radius of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} 2^n x^n$.
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$R = \frac{1}{2}$. Rewrite as $\sum (2x)^n$; radius is $\frac{1}{2}$.
$R = \frac{1}{2}$. Rewrite as $\sum (2x)^n$; radius is $\frac{1}{2}$.
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What does $L < 1$ indicate in the ratio test for a series?
What does $L < 1$ indicate in the ratio test for a series?
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Series converges absolutely. Ratio test shows absolute convergence when limit is less than 1.
Series converges absolutely. Ratio test shows absolute convergence when limit is less than 1.
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What is the radius of convergence when $L = 1$ in the ratio test?
What is the radius of convergence when $L = 1$ in the ratio test?
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Test is inconclusive. When $L = 1$, ratio test gives no information about convergence.
Test is inconclusive. When $L = 1$, ratio test gives no information about convergence.
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Find the radius of convergence for $\textstyle\bigsum_{n=1}^{\text{∞}} \frac{x^n}{n^3}$.
Find the radius of convergence for $\textstyle\bigsum_{n=1}^{\text{∞}} \frac{x^n}{n^3}$.
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$R = \text{∞}$. Cubic decay gives infinite radius via ratio test.
$R = \text{∞}$. Cubic decay gives infinite radius via ratio test.
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State the interval of convergence for a series with $R = 0$.
State the interval of convergence for a series with $R = 0$.
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$\text{x = c}$ only. Zero radius restricts convergence to the center point only.
$\text{x = c}$ only. Zero radius restricts convergence to the center point only.
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What test can be used to determine the convergence of a power series?
What test can be used to determine the convergence of a power series?
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Ratio test or root test. Standard convergence tests for determining series behavior.
Ratio test or root test. Standard convergence tests for determining series behavior.
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What does $L = 1$ imply in the ratio test for convergence?
What does $L = 1$ imply in the ratio test for convergence?
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Test is inconclusive. Ratio test fails when limit equals 1; other methods needed.
Test is inconclusive. Ratio test fails when limit equals 1; other methods needed.
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Determine the interval of convergence for $\sum_{n=1}^{\infty} \frac{x^n}{n}$ using the ratio test.
Determine the interval of convergence for $\sum_{n=1}^{\infty} \frac{x^n}{n}$ using the ratio test.
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$(-1, 1]$. Series $\sum \frac{x^n}{n}$ converges at $x = 1$ by alternating series test.
$(-1, 1]$. Series $\sum \frac{x^n}{n}$ converges at $x = 1$ by alternating series test.
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Which condition indicates absolute convergence of a series?
Which condition indicates absolute convergence of a series?
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If $\textstyle\bigsum |a_n|$ converges. Absolute convergence means the series of absolute values converges.
If $\textstyle\bigsum |a_n|$ converges. Absolute convergence means the series of absolute values converges.
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Find the radius of convergence for $\textstyle\bigsum_{n=1}^{\text{∞}} x^n$ using the root test.
Find the radius of convergence for $\textstyle\bigsum_{n=1}^{\text{∞}} x^n$ using the root test.
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$R = 1$. Root test gives $\lim |x| = |x|$; radius is 1.
$R = 1$. Root test gives $\lim |x| = |x|$; radius is 1.
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Identify the radius of convergence for a geometric series $\sum x^n$.
Identify the radius of convergence for a geometric series $\sum x^n$.
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$R = 1$. Standard geometric series has radius 1.
$R = 1$. Standard geometric series has radius 1.
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State the formula for determining convergence using the root test.
State the formula for determining convergence using the root test.
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$\textstyle \lim_{n \to \infty} \big| a_n \big|^{1/n} < 1$. Root test convergence criterion.
$\textstyle \lim_{n \to \infty} \big| a_n \big|^{1/n} < 1$. Root test convergence criterion.
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Determine the radius of convergence for $\sum_{n=0}^{\infty} \frac{(2x)^n}{n!}$.
Determine the radius of convergence for $\sum_{n=0}^{\infty} \frac{(2x)^n}{n!}$.
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$R = \infty$. Factor out constants; factorial dominates for infinite radius.
$R = \infty$. Factor out constants; factorial dominates for infinite radius.
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What condition implies divergence in the root test?
What condition implies divergence in the root test?
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$\textstyle \lim_{n \to \infty} \big| a_n \big|^{1/n} > 1$. Root test divergence condition.
$\textstyle \lim_{n \to \infty} \big| a_n \big|^{1/n} > 1$. Root test divergence condition.
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Find the radius of convergence of $\textstyle\sum_{n=1}^{\infty} \frac{x^n}{3^n}$.
Find the radius of convergence of $\textstyle\sum_{n=1}^{\infty} \frac{x^n}{3^n}$.
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$R = 3$. Rewrite as $\sum (\frac{x}{3})^n$; geometric series with ratio $\frac{1}{3}$.
$R = 3$. Rewrite as $\sum (\frac{x}{3})^n$; geometric series with ratio $\frac{1}{3}$.
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Identify the convergence interval for $\textstyle\bigsum_{n=0}^{\text{∞}} x^n$ using the ratio test.
Identify the convergence interval for $\textstyle\bigsum_{n=0}^{\text{∞}} x^n$ using the ratio test.
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$(-1, 1)$. Standard geometric series; endpoints both diverge.
$(-1, 1)$. Standard geometric series; endpoints both diverge.
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What is the radius of convergence if $L = \text{∞}$ in the ratio test?
What is the radius of convergence if $L = \text{∞}$ in the ratio test?
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$R = 0$. Infinite limit in ratio test means zero radius.
$R = 0$. Infinite limit in ratio test means zero radius.
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What does it mean if a power series diverges for $x = c + R$?
What does it mean if a power series diverges for $x = c + R$?
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Endpoint $x = c + R$ not in interval of convergence. Endpoint divergence excludes that point from the convergence interval.
Endpoint $x = c + R$ not in interval of convergence. Endpoint divergence excludes that point from the convergence interval.
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Identify the radius of convergence if $R = 0$ for a series.
Identify the radius of convergence if $R = 0$ for a series.
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Series converges only at $x = c$. Zero radius means convergence only at the center point.
Series converges only at $x = c$. Zero radius means convergence only at the center point.
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What does the radius of convergence $R = \text{∞}$ imply?
What does the radius of convergence $R = \text{∞}$ imply?
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Series converges for all $x$. Infinite radius means convergence for every real number.
Series converges for all $x$. Infinite radius means convergence for every real number.
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What is the interval of convergence for a power series?
What is the interval of convergence for a power series?
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Values of $x$ where the series converges. The set of all $x$-values where the power series converges.
Values of $x$ where the series converges. The set of all $x$-values where the power series converges.
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Find the radius of convergence for $a_n = \frac{1}{n!}$ using the ratio test.
Find the radius of convergence for $a_n = \frac{1}{n!}$ using the ratio test.
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$R = \text{∞}$, series converges for all $x$. Factorial growth dominates, making the limit 0, so $R = \frac{1}{0} = \infty$.
$R = \text{∞}$, series converges for all $x$. Factorial growth dominates, making the limit 0, so $R = \frac{1}{0} = \infty$.
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