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AP Calculus BC Flashcards: Representing Series As Power Series

Study Representing Series As Power Series in AP Calculus BC with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

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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.

AP Calculus BC Flashcards: Representing Series As Power Series

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QUESTION

Find the power series representation for 1(1−x)2\frac{1}{(1-x)^2}(1−x)21​.

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ANSWER

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

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Flashcard 1: Find the power series representation for 1(1−x)2\frac{1}{(1-x)^2}(1−x)21​.

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

Flashcard 2: What is the radius of convergence for ∑xn\sum x^n∑xn?

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

Flashcard 3: Identify the power series for 11−2x\frac{1}{1-2x}1−2x1​.

Answer: Sum=(2x)n\text{Sum} = (2x)^nSum=(2x)n for ∣2x∣<1|2x| < 1∣2x∣<1. Geometric series with ratio 2x2x2x requires ∣2x∣<1|2x| < 1∣2x∣<1.

Flashcard 4: Convert Sum=xn3n\text{Sum} = \frac{x^n}{3^n}Sum=3nxn​ into a power series centered at 1.

Answer: Sum=(x−1)n3n\text{Sum} = \frac{(x-1)^n}{3^n}Sum=3n(x−1)n​. Substitute (x−1)(x-1)(x−1) for xxx in the original series.

Flashcard 5: What is the interval of convergence for Sum=xnn2\text{Sum} = \frac{x^n}{n^2}Sum=n2xn​?

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

Flashcard 6: What is the interval of convergence for 1+x+x2+...1 + x + x^2 + \text{...}1+x+x2+...?

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

Flashcard 7: What is the general form of a power series centered at ccc?

Answer: ∑an(x−c)n\sum a_n (x-c)^n∑an​(x−c)n. Standard form where coefficients multiply powers of x−cx - cx−c.

Flashcard 8: What is the power series for 11+x\frac{1}{1+x}1+x1​?

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

Flashcard 9: What is the power series representation for ln(1−x)\text{ln}(1-x)ln(1−x)?

Answer: Sum=−xnn\text{Sum} = -\frac{x^n}{n}Sum=−nxn​ for ∣x∣ <1|x| \text{ \textless 1}∣x∣ <1. Natural log series with −x-x−x substituted for xxx.

Flashcard 10: What is the Taylor series for cos(x)\text{cos}(x)cos(x) centered at c=0c=0c=0?

Answer: Sum=(−1)nx2n(2n)!\text{Sum} = (-1)^n \frac{x^{2n}}{(2n)!}Sum=(−1)n(2n)!x2n​. Maclaurin series for cosine with even powers and alternating signs.

Flashcard 11: State the power series for ln(1+x)\text{ln}(1+x)ln(1+x).

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

Flashcard 12: Determine the radius of convergence for Sum=(2x)n\text{Sum} = (2x)^nSum=(2x)n.

Answer: R=12R = \frac{1}{2}R=21​. Series ∑(2x)n\sum (2x)^n∑(2x)n has radius 12\frac{1}{2}21​ from ratio test.

Flashcard 13: Identify the power series for 11−x\frac{1}{1-x}1−x1​.

Answer: Sum=xn\text{Sum} = x^nSum=xn for ∣x∣<1|x| < 1∣x∣<1. Geometric series formula with first term 1 and ratio xxx.

Flashcard 14: Convert Sum=xn\text{Sum} = x^nSum=xn to a power series centered at c=5c=5c=5.

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

Flashcard 15: What is the Maclaurin series for sin(x)\text{sin}(x)sin(x)?

Answer: Sum=(−1)nx2n+1(2n+1)!\text{Sum} = (-1)^n \frac{x^{2n+1}}{(2n+1)!}Sum=(−1)n(2n+1)!x2n+1​. Maclaurin series for sine with odd powers and alternating signs.

Flashcard 16: Identify the power series for 11−x2\frac{1}{1-x^2}1−x21​.

Answer: Sum=x2n\text{Sum} = x^{2n}Sum=x2n for ∣x∣ <1|x| \text{ \textless 1}∣x∣ <1. Geometric series with x2x^2x2 substitution for even powers only.

Flashcard 17: Determine the power series for e−xe^{-x}e−x.

Answer: Sum=(−1)nxnn!\text{Sum} = (-1)^n \frac{x^n}{n!}Sum=(−1)nn!xn​. Exponential series with −x-x−x substituted for xxx.

Flashcard 18: Find the radius of convergence for Sum=xnn!\text{Sum} = \frac{x^n}{n!}Sum=n!xn​.

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

Flashcard 19: Convert the series Sum=xn4n\text{Sum} = \frac{x^n}{4^n}Sum=4nxn​ to a power series centered at 2.

Answer: Sum=(x−2)n4n\text{Sum} = \frac{(x-2)^n}{4^n}Sum=4n(x−2)n​. Substitute (x−2)(x-2)(x−2) for xxx in series ∑xn4n\sum \frac{x^n}{4^n}∑4nxn​.

Flashcard 20: What is the Taylor series for exe^xex centered at c=0c=0c=0?

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

Flashcard 21: State the formula for the interval of convergence of a power series.

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

Flashcard 22: State the formula for the interval of convergence of a power series.

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

Flashcard 23: What is the radius of convergence for Sum=xn\text{Sum} = x^nSum=xn?

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

Flashcard 24: Identify the power series for 11+x2\frac{1}{1+x^2}1+x21​.

Answer: Sum=(−1)nx2n\text{Sum} = (-1)^n x^{2n}Sum=(−1)nx2n for ∣x∣ <1|x| \text{ \textless 1}∣x∣ <1. Geometric series with x2x^2x2 and alternating signs.

Flashcard 25: Identify the power series for 11−x\frac{1}{1-x}1−x1​.

Answer: Sum=xn\text{Sum} = x^nSum=xn for ∣x∣<1|x| < 1∣x∣<1. Geometric series formula with first term 1 and ratio xxx.

Flashcard 26: What is the Taylor series for cos(x)\text{cos}(x)cos(x) centered at c=0c=0c=0?

Answer: Sum=(−1)nx2n(2n)!\text{Sum} = (-1)^n \frac{x^{2n}}{(2n)!}Sum=(−1)n(2n)!x2n​. Maclaurin series for cosine with even powers and alternating signs.

Flashcard 27: Convert Sum=xn\text{Sum} = x^nSum=xn to a power series centered at c=5c=5c=5.

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

Flashcard 28: Determine the power series for e−xe^{-x}e−x.

Answer: Sum=(−1)nxnn!\text{Sum} = (-1)^n \frac{x^n}{n!}Sum=(−1)nn!xn​. Exponential series with −x-x−x substituted for xxx.

Flashcard 29: Convert the series Sum=xn4n\text{Sum} = \frac{x^n}{4^n}Sum=4nxn​ to a power series centered at 2.

Answer: Sum=(x−2)n4n\text{Sum} = \frac{(x-2)^n}{4^n}Sum=4n(x−2)n​. Substitute (x−2)(x-2)(x−2) for xxx in series ∑xn4n\sum \frac{x^n}{4^n}∑4nxn​.

Flashcard 30: What is the Taylor series for exe^xex centered at c=0c=0c=0?

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