AP Calculus BC Flashcards: Power Series Radius Interval Of Convergence

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

AP Calculus BC

Power Series Radius Interval Of Convergence

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QUESTION

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Determine the interval of convergence for $\textstyle\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n}$ .

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ANSWER

$(-1, 1]$ . Alternating series converges at $x = 1$ ; diverges at $x = -1$ .

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What this deck covers

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

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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.

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Flashcard 1: Determine the interval of convergence for $\textstyle\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n}$ .

Answer: $(-1, 1]$ . Alternating series converges at $x = 1$ ; diverges at $x = -1$ .

Flashcard 2: What is the interval of convergence for a power series?

Answer: Values of $x$ where the series converges. The set of all $x$ -values where the power series converges.

Flashcard 3: What does the radius of convergence $R = \text{∞}$ imply?

Answer: Series converges for all $x$ . Infinite radius means convergence for every real number.

Flashcard 4: What is the radius of convergence when $L = 1$ in the ratio test?

Answer: Test is inconclusive. When $L = 1$ , ratio test gives no information about convergence.

Flashcard 5: State the formula for the root test applied to a series.

Answer: $\textstyle\lim_{n \to \infty} \big|a_n\big|^{1/n} < 1$ for convergence. Root test condition for series convergence.

Flashcard 6: For which $x$ does the series $\textstyle\sum_{n=0}^{\infty} a_n x^n$ converge absolutely?

Answer: $|x| < R$ . Within radius of convergence, series converges absolutely.

Flashcard 7: Identify the radius of convergence for $\sum_{n=0}^{\infty} 2^n x^n$ .

Answer: $R = \frac{1}{2}$ . Rewrite as $\sum (2x)^n$ ; radius is $\frac{1}{2}$ .

Flashcard 8: Determine the radius of convergence for $\sum_{n=0}^{\infty} \frac{(2x)^n}{n!}$ .

Answer: $R = \infty$ . Factor out constants; factorial dominates for infinite radius.

Flashcard 9: What does it mean if a power series diverges for $x = c + R$ ?

Answer: Endpoint $x = c + R$ not in interval of convergence. Endpoint divergence excludes that point from the convergence interval.

Flashcard 10: Identify the radius of convergence for $a_n = \frac{1}{n^2}$ using the ratio test.

Answer: $R = \text{∞}$ . Quadratic decay gives infinite radius via ratio test.

Flashcard 11: Identify the radius of convergence if $R = 0$ for a series.

Answer: Series converges only at $x = c$ . Zero radius means convergence only at the center point.

Flashcard 12: Identify the radius of convergence for a geometric series $\sum x^n$ .

Answer: $R = 1$ . Standard geometric series has radius 1.

Flashcard 13: Find the radius of convergence for $\textstyle\bigsum_{n=1}^{\text{∞}} x^n$ using the root test.

Answer: $R = 1$ . Root test gives $\lim |x| = |x|$ ; radius is 1.

Flashcard 14: Determine the interval of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} \frac{(x-3)^n}{2^n}$ .

Answer: $(1, 5)$ . Series $\sum \frac{(x-3)^n}{2^n}$ has center 3, radius 2; endpoints diverge.

Flashcard 15: What is the radius of convergence if $L = 0$ in the ratio test?

Answer: $R = \text{∞}$ . When limit is 0, radius becomes infinite.

Flashcard 16: Identify the radius of convergence for $\textstyle \sum_{n=0}^{\infty} 5^n x^n$ .

Answer: $R = \frac{1}{5}$ . Rewrite as $\sum (5x)^n$ ; radius is $\frac{1}{5}$ .

Flashcard 17: What test can be used to determine the convergence of a power series?

Answer: Ratio test or root test. Standard convergence tests for determining series behavior.

Flashcard 18: Which condition indicates absolute convergence of a series?

Answer: If $\textstyle\bigsum |a_n|$ converges. Absolute convergence means the series of absolute values converges.

Flashcard 19: Find the radius of convergence for $\textstyle\bigsum_{n=1}^{\text{∞}} \frac{x^n}{n^3}$ .

Answer: $R = \text{∞}$ . Cubic decay gives infinite radius via ratio test.

Flashcard 20: State the formula for the root test applied to a series.

Answer: $\lim_{n \to \infty} \big| a_n \big|^{1/n} < 1$ for convergence. Root test condition for series convergence.

Flashcard 21: Find the radius of convergence for $\textstyle\bigsum_{n=1}^{\text{∞}} \frac{x^n}{n^3}$ .

Answer: $R = \text{∞}$ . Cubic decay gives infinite radius via ratio test.

Flashcard 22: Find the interval of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} \frac{x^n}{n!}$ .

Answer: $(-\text{∞}, \text{∞})$ . Exponential function series; factorial dominates, giving infinite radius.

Flashcard 23: Determine the radius of convergence for $\sum_{n=0}^{\infty} \frac{(2x)^n}{n!}$ .

Answer: $R = \infty$ . Factor out constants; factorial dominates for infinite radius.

Flashcard 24: Identify the convergence interval for $\textstyle\bigsum_{n=0}^{\text{∞}} x^n$ using the ratio test.

Answer: $(-1, 1)$ . Standard geometric series; endpoints both diverge.

Flashcard 25: What test can be used to determine the convergence of a power series?

Answer: Ratio test or root test. Standard convergence tests for determining series behavior.

Flashcard 26: What is the interval of convergence for $\textstyle \sum_{n=1}^{\infty} \frac{x^n}{n^2}$ ?

Answer: $(-\infty, \infty)$ . $p$ -series with $p = 2 > 1$ converges; infinite radius for power series.

Flashcard 27: For which $x$ does the series $\sum_{n=0}^{\infty} a_n x^n$ converge absolutely?

Answer: $|x| < R$ . Within radius of convergence, series converges absolutely.

Flashcard 28: Find the radius of convergence for $a_n = n^2$ using the root test.

Answer: $R = 0$ , series converges only at $x = c$ . Quadratic growth makes $\lim |a_n|^{1/n} = \infty$ , so $R = 0$ .

Flashcard 29: Determine the convergence interval of $\textstyle\sum_{n=1}^{\infty} \frac{x^n}{n}$ .

Answer: $(-1, 1]$ . Alternating harmonic-type series; converges at $x = 1$ by alternating series test.

Flashcard 30: Which condition indicates absolute convergence of a series?

Answer: If $\sum |a_n|$ converges. Absolute convergence means the series of absolute values converges.

Flashcard 31: Determine the interval of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} \frac{(x-3)^n}{2^n}$ .

Answer: $(1, 5)$ . Series $\sum \frac{(x-3)^n}{2^n}$ has center 3, radius 2; endpoints diverge.

Flashcard 32: What does it mean if a power series diverges for $x = c + R$ ?

Answer: Endpoint $x = c + R$ not in interval of convergence. Endpoint divergence excludes that point from the convergence interval.

Flashcard 33: What is the general form of a power series centered at $c$ ?

Answer: $\text{Series: } \textstyle\bigsum_{n=0}^{\text{∞}} a_n(x - c)^n$ . Standard power series form with center $c$ and coefficients $a_n$ .

Flashcard 34: Find the radius of convergence of $\textstyle\sum_{n=1}^{\infty} \frac{x^n}{3^n}$ .

Answer: $R = 3$ . Rewrite as $\sum (\frac{x}{3})^n$ ; geometric series with ratio $\frac{1}{3}$ .

Flashcard 35: Determine the convergence interval of $\textstyle\bigsum_{n=1}^{\infty} \frac{x^n}{n}$

Answer: $(-1, 1]$ . Alternating harmonic-type series; converges at $x = 1$ by alternating series test.

Flashcard 36: Find the interval of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} \frac{x^n}{n!}$ .

Answer: $(-\text{∞}, \text{∞})$ . Exponential function series; factorial dominates, giving infinite radius.

Flashcard 37: State the interval of convergence for a series with $R = 0$ .

Answer: $\text{x = c}$ only. Zero radius restricts convergence to the center point only.

Flashcard 38: What does $L < 1$ indicate in the ratio test for a series?

Answer: Series converges absolutely. Ratio test shows absolute convergence when limit is less than 1.

Flashcard 39: Identify the convergence interval for $\textstyle\bigsum_{n=0}^{\text{∞}} x^n$ using the ratio test.

Answer: $(-1, 1)$ . Standard geometric series; endpoints both diverge.

Flashcard 40: Which test confirms divergence if the limit is greater than 1?

Answer: Ratio test or root test. Both tests indicate divergence when limit exceeds 1.

Flashcard 41: Find the radius of convergence for $\textstyle\bigsum_{n=1}^{\text{∞}} x^n$ using the root test.

Answer: $R = 1$ . Root test gives $\lim |x| = |x|$ ; radius is 1.

Flashcard 42: Identify the radius of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} 5^n x^n$ .

Answer: $R = \frac{1}{5}$ . Rewrite as $\sum (5x)^n$ ; radius is $\frac{1}{5}$ .

Flashcard 43: State the formula for determining convergence using the root test.

Answer: $\textstyle \lim_{n \to \infty} \big| a_n \big|^{1/n} < 1$ . Root test convergence criterion.

Flashcard 44: What is the radius of convergence when $L = 1$ in the ratio test?

Answer: Test is inconclusive. When $L = 1$ , ratio test gives no information about convergence.

Flashcard 45: Find the radius of convergence for $a_n = \frac{1}{n!}$ using the ratio test.

Answer: $R = \infty$ , series converges for all $x$ . Factorial growth dominates, making the limit 0, so $R = \frac{1}{0} = \infty$ .

Flashcard 46: What is the interval of convergence for $\textstyle \sum_{n=0}^{\infty} \frac{x^n}{n!}$ ?

Answer: $(- \infty, \infty)$ . Exponential function has infinite radius of convergence.

Flashcard 47: Determine the interval of convergence for the series $\textstyle \sum_{n=0}^{\infty} x^n$ .

Answer: $-1, 1$ . Geometric series with ratio $|x| < 1$ ; endpoints diverge.

Flashcard 48: What does $L = 1$ imply in the ratio test for convergence?

Answer: Test is inconclusive. Ratio test fails when limit equals 1; other methods needed.

Flashcard 49: State the interval of convergence for a series with $R = 0$ .

Answer: $\text{x = c}$ only. Zero radius restricts convergence to the center point only.

Flashcard 50: What is the radius of convergence if $L = \text{∞}$ in the ratio test?

Answer: $R = 0$ . Infinite limit in ratio test means zero radius.

Flashcard 51: What is the interval of convergence for $\textstyle\bigsum_{n=1}^{\text{∞}} \frac{x^n}{n^2}$ ?

Answer: $(-\text{∞}, \text{∞})$ . $p$ -series with $p = 2 > 1$ converges; infinite radius for power series.

Flashcard 52: What does the radius of convergence $R = \text{∞}$ imply?

Answer: Series converges for all $x$ . Infinite radius means convergence for every real number.

Flashcard 53: Determine the interval of convergence for $\textstyle\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n}$

Answer: $(-1, 1]$ . Alternating series converges at $x = 1$ ; diverges at $x = -1$ .

Flashcard 54: Identify the radius of convergence if $R = 0$ for a series.

Answer: Series converges only at $x = c$ . Zero radius means convergence only at the center point.

Flashcard 55: What does $L < 1$ indicate in the ratio test for a series?

Answer: Series converges absolutely. Ratio test shows absolute convergence when limit is less than 1.

Flashcard 56: What is the general form of a power series centered at $c$ ?

Answer: $\text{Series: } \textstyle\sum_{n=0}^{\infty} a_n(x - c)^n$ . Standard power series form with center $c$ and coefficients $a_n$ .

Flashcard 57: Determine the interval of convergence for $\sum_{n=1}^{\infty} \frac{x^n}{n}$ using the ratio test.

Answer: $(-1, 1]$ . Series $\sum \frac{x^n}{n}$ converges at $x = 1$ by alternating series test.

Flashcard 58: State the formula for the radius of convergence using the ratio test.

Answer: $R = \frac{1}{L}$ where $L = \lim_{n \to \infty} \bigg| \frac{a_{n+1}}{a_n} \bigg|$ . Ratio test formula where $L$ is the limit of consecutive coefficient ratios.

Flashcard 59: Identify the radius of convergence for $\textstyle\bigsum_{n=0}^{\text{∞}} 2^n x^n$ .

Answer: $R = \frac{1}{2}$ . Rewrite as $\sum (2x)^n$ ; radius is $\frac{1}{2}$ .

Flashcard 60: Identify the radius of convergence for a geometric series $\textstyle\bigsum x^n$ .

Answer: $R = 1$ . Standard geometric series has radius 1.

Flashcard 61: What is the interval of convergence for $\textstyle\sum_{n=0}^{\infty} \frac{x^n}{n!}$ ?

Answer: $(-\infty, \infty)$ . Exponential function has infinite radius of convergence.

Flashcard 62: What does $L = 1$ imply in the ratio test for convergence?

Answer: Test is inconclusive. Ratio test fails when limit equals 1; other methods needed.

Flashcard 63: What is the radius of convergence if $L = \text{∞}$ in the ratio test?

Answer: $R = 0$ . Infinite limit in ratio test means zero radius.

Flashcard 64: What is the radius of convergence if $L = 0$ in the ratio test?

Answer: $R = \text{∞}$ . When limit is 0, radius becomes infinite.

Flashcard 65: What is the interval of convergence for a power series?

Answer: Values of $x$ where the series converges. The set of all $x$ -values where the power series converges.

Flashcard 66: What condition implies divergence in the root test?

Answer: $\textstyle \lim_{n \to \infty} \big| a_n \big|^{1/n} > 1$ . Root test divergence condition.

Flashcard 67: Find the radius of convergence for $a_n = \frac{1}{n!}$ using the ratio test.

Answer: $R = \text{∞}$ , series converges for all $x$ . Factorial growth dominates, making the limit 0, so $R = \frac{1}{0} = \infty$ .

Flashcard 68: State the formula for the radius of convergence using the ratio test.

Answer: $R = \frac{1}{L}$ where $L = \textstyle\lim_{n \to \infty} \bigg| \frac{a_{n+1}}{a_n} \bigg|$ . Ratio test formula where $L$ is the limit of consecutive coefficient ratios.

Flashcard 69: Determine the interval of convergence for the series $\sum_{n=0}^{\infty} x^n$ .

Answer: $(-1, 1)$ . Geometric series with ratio $|x| < 1$ ; endpoints diverge.

Flashcard 70: What condition implies divergence in the root test?

Answer: $\lim_{n \to \infty} \big| a_n \big|^{1/n} > 1$ . Root test divergence condition.

Flashcard 71: Determine the interval of convergence for $\textstyle\sum_{n=1}^{\infty} \frac{x^n}{n}$ using the ratio test.

Answer: $(-1, 1]$ . Series $\sum \frac{x^n}{n}$ converges at $x = 1$ by alternating series test.

Flashcard 72: State the formula for determining convergence using the root test.

Answer: $\textstyle\lim_{n \to \infty} \big| a_n \big|^{1/n} < 1$ . Root test convergence criterion.

AP Calculus BC Flashcards: Power Series Radius Interval Of Convergence

Power Series Radius Interval Of Convergence

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Flashcard 1: Determine the interval of convergence for ∑n=1∞(−1)nxnn\textstyle\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n}∑n=1∞​n(−1)nxn​.

Flashcard 2: What is the interval of convergence for a power series?

Flashcard 3: What does the radius of convergence R=∞R = \text{∞}R=∞ imply?

Flashcard 4: What is the radius of convergence when L=1L = 1L=1 in the ratio test?

Flashcard 5: State the formula for the root test applied to a series.

Flashcard 6: For which xxx does the series ∑n=0∞anxn\textstyle\sum_{n=0}^{\infty} a_n x^n∑n=0∞​an​xn converge absolutely?

Flashcard 7: Identify the radius of convergence for ∑n=0∞2nxn\sum_{n=0}^{\infty} 2^n x^n∑n=0∞​2nxn.

Flashcard 8: Determine the radius of convergence for ∑n=0∞(2x)nn!\sum_{n=0}^{\infty} \frac{(2x)^n}{n!}∑n=0∞​n!(2x)n​.

Flashcard 9: What does it mean if a power series diverges for x=c+Rx = c + Rx=c+R?

Flashcard 10: Identify the radius of convergence for an=1n2a_n = \frac{1}{n^2}an​=n21​ using the ratio test.

Flashcard 11: Identify the radius of convergence if R=0R = 0R=0 for a series.

Flashcard 12: Identify the radius of convergence for a geometric series ∑xn\sum x^n∑xn.

Flashcard 13: Find the radius of convergence for \bigsumn=1∞xn\textstyle\bigsum_{n=1}^{\text{∞}} x^n\bigsumn=1∞​xn using the root test.

Flashcard 14: Determine the interval of convergence for \bigsumn=0∞(x−3)n2n\textstyle\bigsum_{n=0}^{\text{∞}} \frac{(x-3)^n}{2^n}\bigsumn=0∞​2n(x−3)n​.

Flashcard 15: What is the radius of convergence if L=0L = 0L=0 in the ratio test?

Flashcard 16: Identify the radius of convergence for ∑n=0∞5nxn\textstyle \sum_{n=0}^{\infty} 5^n x^n∑n=0∞​5nxn.

Flashcard 17: What test can be used to determine the convergence of a power series?

Flashcard 18: Which condition indicates absolute convergence of a series?

Flashcard 19: Find the radius of convergence for \bigsumn=1∞xnn3\textstyle\bigsum_{n=1}^{\text{∞}} \frac{x^n}{n^3}\bigsumn=1∞​n3xn​.

Flashcard 20: State the formula for the root test applied to a series.

Flashcard 21: Find the radius of convergence for \bigsumn=1∞xnn3\textstyle\bigsum_{n=1}^{\text{∞}} \frac{x^n}{n^3}\bigsumn=1∞​n3xn​.

Flashcard 22: Find the interval of convergence for \bigsumn=0∞xnn!\textstyle\bigsum_{n=0}^{\text{∞}} \frac{x^n}{n!}\bigsumn=0∞​n!xn​.

Flashcard 23: Determine the radius of convergence for ∑n=0∞(2x)nn!\sum_{n=0}^{\infty} \frac{(2x)^n}{n!}∑n=0∞​n!(2x)n​.

Flashcard 24: Identify the convergence interval for \bigsumn=0∞xn\textstyle\bigsum_{n=0}^{\text{∞}} x^n\bigsumn=0∞​xn using the ratio test.

Flashcard 25: What test can be used to determine the convergence of a power series?

Flashcard 26: What is the interval of convergence for ∑n=1∞xnn2\textstyle \sum_{n=1}^{\infty} \frac{x^n}{n^2}∑n=1∞​n2xn​?

Flashcard 27: For which xxx does the series ∑n=0∞anxn\sum_{n=0}^{\infty} a_n x^n∑n=0∞​an​xn converge absolutely?

Flashcard 28: Find the radius of convergence for an=n2a_n = n^2an​=n2 using the root test.

Flashcard 29: Determine the convergence interval of ∑n=1∞xnn\textstyle\sum_{n=1}^{\infty} \frac{x^n}{n}∑n=1∞​nxn​.

Flashcard 30: Which condition indicates absolute convergence of a series?

Flashcard 31: Determine the interval of convergence for \bigsumn=0∞(x−3)n2n\textstyle\bigsum_{n=0}^{\text{∞}} \frac{(x-3)^n}{2^n}\bigsumn=0∞​2n(x−3)n​.

Flashcard 32: What does it mean if a power series diverges for x=c+Rx = c + Rx=c+R?

Flashcard 33: What is the general form of a power series centered at ccc?

Flashcard 34: Find the radius of convergence of ∑n=1∞xn3n\textstyle\sum_{n=1}^{\infty} \frac{x^n}{3^n}∑n=1∞​3nxn​.

Flashcard 35: Determine the convergence interval of \bigsumn=1∞xnn\textstyle\bigsum_{n=1}^{\infty} \frac{x^n}{n}\bigsumn=1∞​nxn​

Flashcard 36: Find the interval of convergence for \bigsumn=0∞xnn!\textstyle\bigsum_{n=0}^{\text{∞}} \frac{x^n}{n!}\bigsumn=0∞​n!xn​.

Flashcard 37: State the interval of convergence for a series with R=0R = 0R=0.

Flashcard 38: What does L<1L < 1L<1 indicate in the ratio test for a series?

Flashcard 39: Identify the convergence interval for \bigsumn=0∞xn\textstyle\bigsum_{n=0}^{\text{∞}} x^n\bigsumn=0∞​xn using the ratio test.