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: Which test is most suitable for factorial expressions?

Answer: The Ratio Test. Effective for factorial terms.

Flashcard 2: What does the Root Test involve calculating?

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

Flashcard 3: 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 4: What is the first step in the Alternating Series Test?

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

Flashcard 5: Determine convergence: $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$.

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

Flashcard 6: Apply the Ratio Test: $\sum_{n=1}^{\infty} \frac{n!}{(2n)!}$.

Answer: Converges absolutely. Factorials grow faster than exponentials.

Flashcard 7: Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n}$.

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

Flashcard 8: 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 9: Determine convergence: $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$.

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

Flashcard 10: What test determines absolute convergence using absolute values?

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

Flashcard 11: What characterizes a geometric series?

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

Flashcard 12: Determine convergence: $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$.

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

Flashcard 13: 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 14: 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 15: 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 16: What test can determine if a series converges conditionally?

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

Flashcard 17: 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 18: What is the series $\sum_{n=1}^{\infty} \frac{1}{n}$ known as?

Answer: Harmonic Series. Famous divergent series.

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

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

Flashcard 20: Determine convergence: $\sum_{n=1}^{\infty} n^2 e^{-n}$.

Answer: Converges absolutely. Exponential decay dominates polynomial growth.

Flashcard 21: 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 22: Identify the series: $\sum_{n=0}^{\infty} x^n$ for $|x| < 1$.

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

Flashcard 23: Apply the Root Test: $\sum_{n=1}^{\infty} \left(\frac{n}{n+1}\right)^n$.

Answer: Diverges. Root approaches 1, so diverges.

Flashcard 24: 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 25: Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n^{0.5}}$.

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

Flashcard 26: Which test is most suitable for factorial expressions?

Answer: The Ratio Test. Effective for factorial terms.

Flashcard 27: Identify the series: $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$.

Answer: Alternating Harmonic Series. Classic conditionally convergent series.

Flashcard 28: 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 29: 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 30: What test compares series to determine absolute convergence?

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