Representing Series as Power Series - AP Calculus BC
Card 1 of 30
Find the power series representation for $\frac{1}{(1-x)^2}$.
Find the power series representation for $\frac{1}{(1-x)^2}$.
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$\text{Sum} = (n+1)x^n$ for $|x| < 1$. Derivative of geometric series gives powers with coefficients.
$\text{Sum} = (n+1)x^n$ for $|x| < 1$. Derivative of geometric series gives powers with coefficients.
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What is the radius of convergence for $\sum x^n$?
What is the radius of convergence for $\sum x^n$?
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$R = 1$. Geometric series $\sum x^n$ has unit radius.
$R = 1$. Geometric series $\sum x^n$ has unit radius.
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Identify the power series for $\frac{1}{1-2x}$.
Identify the power series for $\frac{1}{1-2x}$.
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$\text{Sum} = (2x)^n$ for $|2x| < 1$. Geometric series with ratio $2x$ requires $|2x| < 1$.
$\text{Sum} = (2x)^n$ for $|2x| < 1$. Geometric series with ratio $2x$ requires $|2x| < 1$.
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Convert $\text{Sum} = \frac{x^n}{3^n}$ into a power series centered at 1.
Convert $\text{Sum} = \frac{x^n}{3^n}$ into a power series centered at 1.
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$\text{Sum} = \frac{(x-1)^n}{3^n}$. Substitute $(x-1)$ for $x$ in the original series.
$\text{Sum} = \frac{(x-1)^n}{3^n}$. Substitute $(x-1)$ for $x$ in the original series.
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What is the interval of convergence for $\text{Sum} = \frac{x^n}{n^2}$?
What is the interval of convergence for $\text{Sum} = \frac{x^n}{n^2}$?
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$|x| \text{ \textless 1}$. P-series with $p=2>1$ converges when $|x| < 1$.
$|x| \text{ \textless 1}$. P-series with $p=2>1$ converges when $|x| < 1$.
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What is the interval of convergence for $1 + x + x^2 + \text{...}$?
What is the interval of convergence for $1 + x + x^2 + \text{...}$?
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$|x| \text{ \textless 1}$. Basic geometric series convergence condition.
$|x| \text{ \textless 1}$. Basic geometric series convergence condition.
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What is the general form of a power series centered at $c$?
What is the general form of a power series centered at $c$?
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$\sum a_n (x-c)^n$. Standard form where coefficients multiply powers of $x - c$.
$\sum a_n (x-c)^n$. Standard form where coefficients multiply powers of $x - c$.
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What is the power series for $\frac{1}{1+x}$?
What is the power series for $\frac{1}{1+x}$?
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$\text{Sum} = (-1)^n x^n$ for $|x| < 1$. Geometric series with alternating signs from ratio $-x$.
$\text{Sum} = (-1)^n x^n$ for $|x| < 1$. Geometric series with alternating signs from ratio $-x$.
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What is the power series representation for $\text{ln}(1-x)$?
What is the power series representation for $\text{ln}(1-x)$?
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$\text{Sum} = -\frac{x^n}{n}$ for $|x| \text{ \textless 1}$. Natural log series with $-x$ substituted for $x$.
$\text{Sum} = -\frac{x^n}{n}$ for $|x| \text{ \textless 1}$. Natural log series with $-x$ substituted for $x$.
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What is the Taylor series for $\text{cos}(x)$ centered at $c=0$?
What is the Taylor series for $\text{cos}(x)$ centered at $c=0$?
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$\text{Sum} = (-1)^n \frac{x^{2n}}{(2n)!}$. Maclaurin series for cosine with even powers and alternating signs.
$\text{Sum} = (-1)^n \frac{x^{2n}}{(2n)!}$. Maclaurin series for cosine with even powers and alternating signs.
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State the power series for $\text{ln}(1+x)$.
State the power series for $\text{ln}(1+x)$.
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$\text{Sum} = (-1)^{n+1} \frac{x^n}{n}$ for $|x| \text{ \textless 1}$. Alternating harmonic series with convergence $|x| < 1$.
$\text{Sum} = (-1)^{n+1} \frac{x^n}{n}$ for $|x| \text{ \textless 1}$. Alternating harmonic series with convergence $|x| < 1$.
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Determine the radius of convergence for $\text{Sum} = (2x)^n$.
Determine the radius of convergence for $\text{Sum} = (2x)^n$.
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$R = \frac{1}{2}$. Series $\sum (2x)^n$ has radius $\frac{1}{2}$ from ratio test.
$R = \frac{1}{2}$. Series $\sum (2x)^n$ has radius $\frac{1}{2}$ from ratio test.
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Identify the power series for $\frac{1}{1-x}$.
Identify the power series for $\frac{1}{1-x}$.
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$\text{Sum} = x^n$ for $|x| < 1$. Geometric series formula with first term 1 and ratio $x$.
$\text{Sum} = x^n$ for $|x| < 1$. Geometric series formula with first term 1 and ratio $x$.
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Convert $\text{Sum} = x^n$ to a power series centered at $c=5$.
Convert $\text{Sum} = x^n$ to a power series centered at $c=5$.
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$\text{Sum} = (x-5)^n$. Shift power series by substituting $(x-5)$ for $x$.
$\text{Sum} = (x-5)^n$. Shift power series by substituting $(x-5)$ for $x$.
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What is the Maclaurin series for $\text{sin}(x)$?
What is the Maclaurin series for $\text{sin}(x)$?
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$\text{Sum} = (-1)^n \frac{x^{2n+1}}{(2n+1)!}$. Maclaurin series for sine with odd powers and alternating signs.
$\text{Sum} = (-1)^n \frac{x^{2n+1}}{(2n+1)!}$. Maclaurin series for sine with odd powers and alternating signs.
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Identify the power series for $\frac{1}{1-x^2}$.
Identify the power series for $\frac{1}{1-x^2}$.
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$\text{Sum} = x^{2n}$ for $|x| \text{ \textless 1}$. Geometric series with $x^2$ substitution for even powers only.
$\text{Sum} = x^{2n}$ for $|x| \text{ \textless 1}$. Geometric series with $x^2$ substitution for even powers only.
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Determine the power series for $e^{-x}$.
Determine the power series for $e^{-x}$.
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$\text{Sum} = (-1)^n \frac{x^n}{n!}$. Exponential series with $-x$ substituted for $x$.
$\text{Sum} = (-1)^n \frac{x^n}{n!}$. Exponential series with $-x$ substituted for $x$.
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Find the radius of convergence for $\text{Sum} = \frac{x^n}{n!}$.
Find the radius of convergence for $\text{Sum} = \frac{x^n}{n!}$.
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$R = \text{infinity}$. Exponential series converges for all real values.
$R = \text{infinity}$. Exponential series converges for all real values.
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Convert the series $\text{Sum} = \frac{x^n}{4^n}$ to a power series centered at 2.
Convert the series $\text{Sum} = \frac{x^n}{4^n}$ to a power series centered at 2.
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$\text{Sum} = \frac{(x-2)^n}{4^n}$. Substitute $(x-2)$ for $x$ in series $\sum \frac{x^n}{4^n}$.
$\text{Sum} = \frac{(x-2)^n}{4^n}$. Substitute $(x-2)$ for $x$ in series $\sum \frac{x^n}{4^n}$.
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What is the Taylor series for $e^x$ centered at $c=0$?
What is the Taylor series for $e^x$ centered at $c=0$?
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$\text{Sum} = \frac{x^n}{n!}$. Maclaurin series for exponential function at origin.
$\text{Sum} = \frac{x^n}{n!}$. Maclaurin series for exponential function at origin.
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State the formula for the interval of convergence of a power series.
State the formula for the interval of convergence of a power series.
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$|x-c| < R$. Distance from center must be less than radius of convergence.
$|x-c| < R$. Distance from center must be less than radius of convergence.
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State the formula for the interval of convergence of a power series.
State the formula for the interval of convergence of a power series.
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$|x-c| < R$. Distance from center must be less than radius of convergence.
$|x-c| < R$. Distance from center must be less than radius of convergence.
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What is the radius of convergence for $\text{Sum} = x^n$?
What is the radius of convergence for $\text{Sum} = x^n$?
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$R = 1$. Geometric series $\sum x^n$ has unit radius.
$R = 1$. Geometric series $\sum x^n$ has unit radius.
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Identify the power series for $\frac{1}{1+x^2}$.
Identify the power series for $\frac{1}{1+x^2}$.
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$\text{Sum} = (-1)^n x^{2n}$ for $|x| \text{ \textless 1}$. Geometric series with $x^2$ and alternating signs.
$\text{Sum} = (-1)^n x^{2n}$ for $|x| \text{ \textless 1}$. Geometric series with $x^2$ and alternating signs.
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Identify the power series for $\frac{1}{1-x}$.
Identify the power series for $\frac{1}{1-x}$.
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$\text{Sum} = x^n$ for $|x| < 1$. Geometric series formula with first term 1 and ratio $x$.
$\text{Sum} = x^n$ for $|x| < 1$. Geometric series formula with first term 1 and ratio $x$.
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What is the Taylor series for $\text{cos}(x)$ centered at $c=0$?
What is the Taylor series for $\text{cos}(x)$ centered at $c=0$?
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$\text{Sum} = (-1)^n \frac{x^{2n}}{(2n)!}$. Maclaurin series for cosine with even powers and alternating signs.
$\text{Sum} = (-1)^n \frac{x^{2n}}{(2n)!}$. Maclaurin series for cosine with even powers and alternating signs.
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Convert $\text{Sum} = x^n$ to a power series centered at $c=5$.
Convert $\text{Sum} = x^n$ to a power series centered at $c=5$.
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$\text{Sum} = (x-5)^n$. Shift power series by substituting $(x-5)$ for $x$.
$\text{Sum} = (x-5)^n$. Shift power series by substituting $(x-5)$ for $x$.
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Determine the power series for $e^{-x}$.
Determine the power series for $e^{-x}$.
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$\text{Sum} = (-1)^n \frac{x^n}{n!}$. Exponential series with $-x$ substituted for $x$.
$\text{Sum} = (-1)^n \frac{x^n}{n!}$. Exponential series with $-x$ substituted for $x$.
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Convert the series $\text{Sum} = \frac{x^n}{4^n}$ to a power series centered at 2.
Convert the series $\text{Sum} = \frac{x^n}{4^n}$ to a power series centered at 2.
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$\text{Sum} = \frac{(x-2)^n}{4^n}$. Substitute $ (x-2) $ for $ x $ in series $ \sum \frac{x^n}{4^n} $.
$\text{Sum} = \frac{(x-2)^n}{4^n}$. Substitute $ (x-2) $ for $ x $ in series $ \sum \frac{x^n}{4^n} $.
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What is the Taylor series for $e^x$ centered at $c=0$?
What is the Taylor series for $e^x$ centered at $c=0$?
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$\text{Sum} = \frac{x^n}{n!}$. Maclaurin series for exponential function at origin.
$\text{Sum} = \frac{x^n}{n!}$. Maclaurin series for exponential function at origin.
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