AP Calculus BC Flashcards: Representing Series As Power Series

What this deck covers

This deck focuses on Representing Series As Power Series, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Convert $\text{Sum} = x^n$ to a power series centered at $c=5$.

Answer: $\text{Sum} = (x-5)^n$. Shift power series by substituting $(x-5)$ for $x$.

Flashcard 2: Find the radius of convergence for $\text{Sum} = \frac{x^n}{n!}$.

Answer: $R = \text{infinity}$. Exponential series converges for all real values.

Flashcard 3: Determine the radius of convergence for $\text{Sum} = (2x)^n$.

Answer: $R = \frac{1}{2}$. Series $\sum (2x)^n$ has radius $\frac{1}{2}$ from ratio test.

Flashcard 4: What is the interval of convergence for $1 + x + x^2 + \text{...}$?

Answer: $|x| \text{ \textless 1}$. Basic geometric series convergence condition.

Flashcard 5: Convert the series $\text{Sum} = \frac{x^n}{n!}$ to a power series centered at 3.

Answer: $\text{Sum} = \frac{(x-3)^n}{n!}$. Shift exponential series by substituting $x-3$ for $x$.

Flashcard 6: Identify the power series for $\frac{1}{1+x^2}$.

Answer: $\text{Sum} = (-1)^n x^{2n}$ for $|x| \text{ \textless 1}$. Geometric series with $x^2$ and alternating signs.

Flashcard 7: Identify the power series for $\frac{1}{1-x}$.

Answer: $\text{Sum} = x^n$ for $|x| < 1$. Geometric series formula with first term 1 and ratio $x$.

Flashcard 8: Convert $\text{Sum} = \frac{x^n}{3^n}$ into a power series centered at 1.

Answer: $\text{Sum} = \frac{(x-1)^n}{3^n}$. Substitute $(x-1)$ for $x$ in the original series.

Flashcard 9: What is the interval of convergence for $\text{Sum} = \frac{x^n}{n^2}$?

Answer: $|x| \text{ \textless 1}$. P-series with $p=2>1$ converges when $|x| < 1$.

Flashcard 10: Convert the series $\text{Sum} = \frac{x^n}{4^n}$ to a power series centered at 2.

Answer: $\text{Sum} = \frac{(x-2)^n}{4^n}$. Substitute $(x-2)$ for $x$ in series $\sum \frac{x^n}{4^n}$.

Flashcard 11: What is the power series for $\frac{1}{1+x}$?

Answer: $\text{Sum} = (-1)^n x^n$ for $|x| < 1$. Geometric series with alternating signs from ratio $-x$.

Flashcard 12: State the formula for the interval of convergence of a power series.

Answer: $|x-c| < R$. Distance from center must be less than radius of convergence.

Flashcard 13: Identify the power series for $\frac{1}{1-2x}$.

Answer: $\text{Sum} = (2x)^n$ for $|2x| < 1$. Geometric series with ratio $2x$ requires $|2x| < 1$.

Flashcard 14: Determine the interval of convergence for $\text{Sum} = \frac{x^n}{n^3}$.

Answer: $|x| < 1$. P-series with $p=3>1$ converges when $|x| < 1$.

Flashcard 15: Identify the power series for $\frac{1}{1-x^2}$.

Answer: $\text{Sum} = x^{2n}$ for $|x| \text{ \textless 1}$. Geometric series with $x^2$ substitution for even powers only.

Flashcard 16: What is the Maclaurin series for $\text{sin}(x)$?

Answer: $\text{Sum} = (-1)^n \frac{x^{2n+1}}{(2n+1)!}$. Maclaurin series for sine with odd powers and alternating signs.

Flashcard 17: Identify the power series for $\frac{1}{1-x^2}$.

Answer: $\text{Sum} = x^{2n}$ for $|x| \text{ \textless 1}$. Geometric series with $x^2$ substitution for even powers only.

Flashcard 18: Determine the interval of convergence for $\text{Sum} = \frac{x^n}{n^3}$.

Answer: $|x| < 1$. P-series with $p=3>1$ converges when $|x| < 1$.

Flashcard 19: Find the interval of convergence for $\text{Sum} = \frac{x^n}{n}$.

Answer: $|x| < 1$. Harmonic-like series converges when $|x| < 1$.

Flashcard 20: What is the general form of a power series centered at $c$?

Answer: $\sum a_n (x-c)^n$. Standard form where coefficients multiply powers of $x - c$.

Flashcard 21: State the power series for $\tan^{-1}(x)$.

Answer: $\text{Sum} = (-1)^n \frac{x^{2n+1}}{2n+1}$ for $|x| \text{ \textless 1}$. Alternating series with odd powers and odd factorial-like denominators.

Flashcard 22: Convert the series $\text{Sum} = \frac{x^n}{4^n}$ to a power series centered at 2.

Answer: $\text{Sum} = \frac{(x-2)^n}{4^n}$. Substitute $(x-2)$ for $x$ in series $\sum \frac{x^n}{4^n}$.

Flashcard 23: What is the interval of convergence for $\text{Sum} = \frac{x^n}{n^2}$?

Answer: $|x| \text{ \textless 1}$. P-series with $p=2>1$ converges when $|x| < 1$.

Flashcard 24: What is the power series representation for $\text{ln}(1-x)$?

Answer: $\text{Sum} = -\frac{x^n}{n}$ for $|x| \text{ \textless 1}$. Natural log series with $-x$ substituted for $x$.

Flashcard 25: State the power series for $\tan^{-1}(x)$.

Answer: $\text{Sum} = (-1)^n \frac{x^{2n+1}}{2n+1}$ for $|x| \text{ \textless 1}$. Alternating series with odd powers and odd factorial-like denominators.

Flashcard 26: Which test can determine the convergence of a power series?

Answer: The Ratio Test. Uses ratios of consecutive terms to find convergence radius.

Flashcard 27: Determine the power series for $e^{-x}$.

Answer: $\text{Sum} = (-1)^n \frac{x^n}{n!}$. Exponential series with $-x$ substituted for $x$.

Flashcard 28: What is the power series for $\frac{1}{1+x}$?

Answer: $\text{Sum} = (-1)^n x^n$ for $|x| < 1$. Geometric series with alternating signs from ratio $-x$.

Flashcard 29: State the formula for the interval of convergence of a power series.

Answer: $|x-c| < R$. Distance from center must be less than radius of convergence.

Flashcard 30: State the power series for $\text{ln}(1+x)$.

Answer: $\text{Sum} = (-1)^{n+1} \frac{x^n}{n}$ for $|x| \text{ \textless 1}$. Alternating harmonic series with convergence $|x| < 1$.

Flashcard 31: Find the radius of convergence for $\text{Sum} = \frac{x^n}{n!}$.

Answer: $R = \text{infinity}$. Exponential series converges for all real values.

Flashcard 32: What is the interval of convergence for $1 + x + x^2 + \text{...}$?

Answer: $|x| \text{ \textless 1}$. Basic geometric series convergence condition.

Flashcard 33: Determine the power series for $e^{-x}$.

Answer: $\text{Sum} = (-1)^n \frac{x^n}{n!}$. Exponential series with $-x$ substituted for $x$.

Flashcard 34: State the power series for $\text{ln}(1+x)$.

Answer: $\text{Sum} = (-1)^{n+1} \frac{x^n}{n}$ for $|x| \text{ \textless 1}$. Alternating harmonic series with convergence $|x| < 1$.

Flashcard 35: What is the Maclaurin series for $\text{sin}(x)$?

Answer: $\text{Sum} = (-1)^n \frac{x^{2n+1}}{(2n+1)!}$. Maclaurin series for sine with odd powers and alternating signs.

Flashcard 36: Find the power series representation for $\frac{1}{(1-x)^2}$.

Answer: $\text{Sum} = (n+1)x^n$ for $|x| < 1$. Derivative of geometric series gives powers with coefficients.

Flashcard 37: What is the Taylor series for $e^x$ centered at $c=0$?

Answer: $\text{Sum} = \frac{x^n}{n!}$. Maclaurin series for exponential function at origin.

Flashcard 38: Convert the series $\text{Sum} = x^n$ to a power series centered at 2.

Answer: $\text{Sum} = (x-2)^n$. Substitute $(x-2)$ for $x$ in original series.

Flashcard 39: What is the general form of a power series centered at $c$?

Answer: $\text{Sum} = \text{a}_n(x-c)^n$. Standard form where coefficients multiply powers of $(x-c)$.

Flashcard 40: What is the radius of convergence for $\sum x^n$?

Answer: $R = 1$. Geometric series $\sum x^n$ has unit radius.

Flashcard 41: Identify the power series for $\frac{1}{1-2x}$.

Answer: $\text{Sum} = (2x)^n$ for $|2x| < 1$. Geometric series with ratio $2x$ requires $|2x| < 1$.

Flashcard 42: Find the interval of convergence for $\text{Sum} = \frac{x^n}{n}$.

Answer: $|x| \text{ \textless 1}$. Harmonic-like series converges when $|x| < 1$.

Flashcard 43: Identify the power series for $\frac{1}{1+x^2}$.

Answer: $\text{Sum} = (-1)^n x^{2n}$ for $|x| \text{ \textless 1}$. Geometric series with $x^2$ and alternating signs.

Flashcard 44: What is the radius of convergence for $\text{Sum} = x^n$?

Answer: $R = 1$. Geometric series $\sum x^n$ has unit radius.

Flashcard 45: What is the Taylor series for $\text{cos}(x)$ centered at $c=0$?

Answer: $\text{Sum} = (-1)^n \frac{x^{2n}}{(2n)!}$. Maclaurin series for cosine with even powers and alternating signs.

Flashcard 46: Find the power series representation for $\frac{1}{(1-x)^2}$.

Answer: $\text{Sum} = (n+1)x^n$ for $|x| < 1$. Derivative of geometric series gives powers with coefficients.

Flashcard 47: Determine the radius of convergence for $\text{Sum} = (2x)^n$.

Answer: $R = \frac{1}{2}$. Series $\sum (2x)^n$ has radius $\frac{1}{2}$ from ratio test.

Flashcard 48: Which test can determine the convergence of a power series?

Answer: The Ratio Test. Uses ratios of consecutive terms to find convergence radius.

Flashcard 49: Convert $\text{Sum} = \frac{x^n}{3^n}$ into a power series centered at 1.

Answer: $\text{Sum} = \frac{(x-1)^n}{3^n}$. Substitute $(x-1)$ for $x$ in the original series.

Flashcard 50: Identify the power series for $\frac{1}{1-x}$.

Answer: $\text{Sum} = x^n$ for $|x| < 1$. Geometric series formula with first term 1 and ratio $x$.

Flashcard 51: What is the Taylor series for $\text{cos}(x)$ centered at $c=0$?

Answer: $\text{Sum} = (-1)^n \frac{x^{2n}}{(2n)!}$. Maclaurin series for cosine with even powers and alternating signs.

Flashcard 52: Convert the series $\text{Sum} = x^n$ to a power series centered at 2.

Answer: $\text{Sum} = (x-2)^n$. Substitute $(x-2)$ for $x$ in original series.

Flashcard 53: What is the Taylor series for $e^x$ centered at $c=0$?

Answer: $\text{Sum} = \frac{x^n}{n!}$. Maclaurin series for exponential function at origin.

Flashcard 54: Convert $\text{Sum} = x^n$ to a power series centered at $c=5$.

Answer: $\text{Sum} = (x-5)^n$. Shift power series by substituting $(x-5)$ for $x$.

Flashcard 55: What is the power series representation for $\text{ln}(1-x)$?

Answer: $\text{Sum} = -\frac{x^n}{n}$ for $|x| \text{ \textless 1}$. Natural log series with $-x$ substituted for $x$.

Flashcard 56: Convert the series $\text{Sum} = \frac{x^n}{n!}$ to a power series centered at 3.

Answer: $\text{Sum} = \frac{(x-3)^n}{n!}$. Shift exponential series by substituting $(x-3)$ for $x$.