AP Calculus BC Flashcards: Determining Absolute Or Conditional Convergence

What this deck covers

This deck focuses on Determining Absolute Or Conditional Convergence, 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: What is a necessary condition for series convergence?

Answer: The terms must approach zero as $n \to \infty$. Required for any convergent series.

Flashcard 2: What characterizes a geometric series?

Answer: Each term is a constant multiple of the previous term. Common ratio between consecutive terms.

Flashcard 3: What is the first step in the Alternating Series Test?

Answer: Check if $a_n$ is decreasing. Monotonic decreasing requirement.

Flashcard 4: State the condition for a series to converge using the Alternating Series Test.

Answer: $a_n$ decreases to $0$ and $a_n > a_{n+1}$. Terms must decrease monotonically to zero.

Flashcard 5: Identify the series: $\sum_{n=0}^{\infty} x^n$ for $|x| < 1$.

Answer: Geometric Series. Series with constant ratio between terms.

Flashcard 6: Determine convergence: $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$.

Answer: Converges conditionally. Passes Alternating Series Test but absolute values diverge.

Flashcard 7: Determine convergence: $\sum_{n=0}^{\infty} x^n$ for $|x| < 1$.

Answer: Converges absolutely. Common ratio satisfies convergence condition.

Flashcard 8: What is required for the Limit Comparison Test?

Answer: A second series $b_n$ to compare with $a_n$. Need comparison series with known behavior.

Flashcard 9: What test compares series to determine absolute convergence?

Answer: The Comparison Test. Compares term by term with known series.

Flashcard 10: Identify the series: $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$.

Answer: Alternating Harmonic Series. Classic conditionally convergent series.

Flashcard 11: What is the series $\sum_{n=1}^{\infty} \frac{1}{n}$ known as?

Answer: Harmonic Series. Famous divergent series.

Flashcard 12: Apply the Root Test: $\sum_{n=1}^{\infty} \left(\frac{n}{n+1}\right)^n$.

Answer: Diverges. Root approaches 1, so diverges.

Flashcard 13: What is the definition of conditional convergence?

Answer: A series converges conditionally if it converges, but not absolutely. Series converges but not when absolute values are taken.

Flashcard 14: Identify the series: $\sum_{n=0}^{\infty} x^n$ for $|x| < 1$.

Answer: Geometric Series. Series with constant ratio between terms.

Flashcard 15: What is the condition for convergence using the Root Test?

Answer: If $\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1$, converges absolutely. nth root of terms approaches value less than 1.

Flashcard 16: What is the definition of absolute convergence?

Answer: A series converges absolutely if the series of absolute values converges. Convergence when taking absolute values.

Flashcard 17: What is a necessary condition for series convergence?

Answer: The terms must approach zero as $n \to \infty$. Required for any convergent series.

Flashcard 18: Determine convergence: $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$.

Answer: Converges conditionally. Alternating p-series with $p=1/2<1$.

Flashcard 19: Determine convergence: $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$.

Answer: Converges absolutely. Alternating p-series with $p=2>1$.

Flashcard 20: Explain the role of absolute values in convergence tests.

Answer: Used to test absolute convergence. Determines if series converges absolutely.

Flashcard 21: Apply the Root Test: $\sum_{n=1}^{\infty} \left(\frac{1}{3}\right)^n$.

Answer: Converges absolutely. Geometric series with ratio $1/3 < 1$.

Flashcard 22: Explain the role of absolute values in convergence tests.

Answer: Used to test absolute convergence. Determines if series converges absolutely.

Flashcard 23: Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n}$.

Answer: Diverges. P-series with $p=1$ diverges.

Flashcard 24: Which test is most suitable for factorial expressions?

Answer: The Ratio Test. Effective for factorial terms.

Flashcard 25: Apply the Root Test: $\sum_{n=1}^{\infty} \left(\frac{n}{n+1}\right)^n$.

Answer: Diverges. Root approaches 1, so diverges.

Flashcard 26: Determine convergence: $\sum_{n=1}^{\infty} n^2 e^{-n}$.

Answer: Converges absolutely. Exponential decay dominates polynomial growth.

Flashcard 27: What is the first step in the Alternating Series Test?

Answer: Check if $a_n$ is decreasing. Monotonic decreasing requirement.

Flashcard 28: Apply the Ratio Test: $\sum_{n=1}^{\infty} \frac{n!}{(2n)!}$.

Answer: Converges absolutely. Factorials grow faster than exponentials.

Flashcard 29: What test determines absolute convergence using absolute values?

Answer: The Absolute Convergence Test. Tests convergence of $\sum |a_n|$.

Flashcard 30: What characterizes a geometric series?

Answer: Each term is a constant multiple of the previous term. Common ratio between consecutive terms.

Flashcard 31: Determine convergence: $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$.

Answer: Converges conditionally. Alternating p-series with $p=1/2<1$.

Flashcard 32: What is the definition of conditional convergence?

Answer: A series converges conditionally if it converges, but not absolutely. Series converges but not when absolute values are taken.

Flashcard 33: Apply the Root Test: $\sum_{n=1}^{\infty} \left(\frac{1}{3}\right)^n$.

Answer: Converges absolutely. Geometric series with ratio $1/3 < 1$.

Flashcard 34: Determine convergence: $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$.

Answer: Converges absolutely. Alternating p-series with $p=2>1$.

Flashcard 35: What is the definition of absolute convergence?

Answer: A series converges absolutely if the series of absolute values converges. Convergence when taking absolute values.

Flashcard 36: Apply the p-series test: $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$.

Answer: Converges absolutely. P-series with $p=3/2>1$ converges.

Flashcard 37: What does the Root Test involve calculating?

Answer: $\lim_{n \to \infty} \sqrt[n]{|a_n|}$. Formula for root test calculation.

Flashcard 38: Identify the limit for convergence in the Ratio Test.

Answer: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$. Formula for ratio test calculation.

Flashcard 39: State the condition for a series to converge using the Alternating Series Test.

Answer: $a_n$ decreases to $0$ and $a_n > a_{n+1}$. Terms must decrease monotonically to zero.

Flashcard 40: Apply the Comparison Test: $\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$.

Answer: Converges absolutely. Compare with convergent p-series $\sum \frac{1}{n^2}$.

Flashcard 41: What test compares series to determine absolute convergence?

Answer: The Comparison Test. Compares term by term with known series.

Flashcard 42: What is the p-series test condition for convergence?

Answer: If $p > 1$, the p-series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges. Exponent determines convergence behavior.

Flashcard 43: What test can determine if a series converges conditionally?

Answer: The Alternating Series Test. For series with alternating signs.

Flashcard 44: What is the p-series test condition for convergence?

Answer: If $p > 1$, the p-series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges. Exponent determines convergence behavior.

Flashcard 45: What does the Root Test involve calculating?

Answer: $\lim_{n \to \infty} \sqrt[n]{|a_n|}$. Formula for root test calculation.

Flashcard 46: Determine convergence: $\sum_{n=0}^{\infty} x^n$ for $|x| < 1$.

Answer: Converges absolutely. Common ratio satisfies convergence condition.

Flashcard 47: What is the condition for convergence using the Root Test?

Answer: If $\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1$, converges absolutely. nth root of terms approaches value less than 1.

Flashcard 48: Apply the Comparison Test: $\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$.

Answer: Converges absolutely. Compare with convergent p-series $\sum \frac{1}{n^2}$.

Flashcard 49: What is required for the Limit Comparison Test?

Answer: A second series $b_n$ to compare with $a_n$. Need comparison series with known behavior.

Flashcard 50: Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n^{0.5}}$.

Answer: Diverges. P-series with $p=0.5<1$ diverges.

Flashcard 51: Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n^2}$.

Answer: Converges absolutely. P-series with $p=2>1$ converges.

Flashcard 52: Apply the p-series test: $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$.

Answer: Converges absolutely. P-series with $p=3/2>1$ converges.

Flashcard 53: Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n}$.

Answer: Diverges. P-series with $p=1$ diverges.

Flashcard 54: Determine convergence: $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$.

Answer: Converges conditionally. Passes Alternating Series Test but absolute values diverge.

Flashcard 55: Identify the limit for convergence in the Ratio Test.

Answer: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$. Formula for ratio test calculation.

Flashcard 56: What is the series $\sum_{n=1}^{\infty} \frac{1}{n}$ known as?

Answer: Harmonic Series. Famous divergent series.

Flashcard 57: What is the Ratio Test's condition for absolute convergence?

Answer: If $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1$, converges absolutely. Ratio of consecutive terms approaches value less than 1.

Flashcard 58: Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n^2}$.

Answer: Converges absolutely. P-series with $p=2>1$ converges.

Flashcard 59: Which test is most suitable for factorial expressions?

Answer: The Ratio Test. Effective for factorial terms.

Flashcard 60: Identify the series: $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$.

Answer: Alternating Harmonic Series. Classic conditionally convergent series.

Flashcard 61: Apply the Ratio Test: $\sum_{n=1}^{\infty} \frac{n!}{(2n)!}$.

Answer: Converges absolutely. Factorials grow faster than exponentials.