AP Calculus BC Flashcards: Alternating Series Test For Convergence

What this deck covers

This deck focuses on Alternating Series Test For 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: Find if the series $\textstyle \sum_{n=1}^{\infty} (-1)^n \frac{1}{n^5}$ converges.

Answer: Yes, it converges by the Alternating Series Test. Higher powers guarantee all alternating series test conditions are satisfied.

Flashcard 2: What is the Alternating Series Estimation Theorem?

Answer: Provides error bound for partial sums of convergent series. It quantifies how close partial sums are to the series sum.

Flashcard 3: Determine whether $\textstyle \sum_{n=1}^{\infty} (-1)^n n$ converges.

Answer: No, the terms do not approach zero. The terms $a_n = n$ grow without bound, violating the limit condition.

Flashcard 4: Does the series $\textstyle\sum_{n=1}^{\infty} (-1)^n (1 + \frac{1}{n^2})$ converge?

Answer: No, it does not converge. The limit $\lim_{n\to\infty} (1 + \frac{1}{n^2}) = 1 \neq 0$.

Flashcard 5: State the Alternating Series Remainder Theorem.

Answer: The remainder is less than the first unused term. This gives an upper bound on the approximation error.

Flashcard 6: Find if $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{\text{ln}(n+1)}$ converges.

Answer: Yes, it converges by the Alternating Series Test. All conditions are met: positive terms, decreasing, limit to zero.

Flashcard 7: Does the series $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{n}$ satisfy $a_n \to 0$?

Answer: Yes, $a_n = \frac{1}{n} \to 0$ as $n \to \text{inf}$. The harmonic terms $\frac{1}{n}$ clearly approach zero as $n \to \infty$.

Flashcard 8: What happens if $a_n$ does not decrease to 0 in an alternating series?

Answer: The series diverges. The series fails to meet the necessary convergence conditions.

Flashcard 9: Identify the series that does not converge: (A) $\textstyle\sum (-1)^n \frac{1}{n^3}$, (B) $\textstyle\sum (-1)^n n$.

Answer: (B) does not converge. Series (B) has unbounded terms while (A) satisfies all conditions.

Flashcard 10: State the limit condition for the Alternating Series Test.

Answer: $\textstyle \lim_{n \to \infty} a_n = 0$. The terms must approach zero for the series to have a chance at convergence.

Flashcard 11: Does the series $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{n^2}$ converge?

Answer: Yes, it converges by the Alternating Series Test. All three conditions are satisfied: positive, decreasing, limit to zero.

Flashcard 12: What does it mean for the terms to be 'eventually decreasing'?

Answer: There exists $N$ such that $a_{n+1} < a_n$ for $n > N$. The decreasing condition only needs to hold for sufficiently large $n$.

Flashcard 13: Determine if $\textstyle\bigsum_{n=1}^{\infty} (-1)^n \frac{1}{n+1}$ satisfies $a_n \to 0$.

Answer: Yes, $a_n = \frac{1}{n+1} \to 0$ as $n \to \infty$. The shifted harmonic terms still approach zero as required.

Flashcard 14: What type of series does the Alternating Series Test apply to?

Answer: Applies to series with terms $(-1)^n a_n$. The alternating factor $(-1)^n$ creates the sign pattern.

Flashcard 15: What does it mean if a series is absolutely convergent?

Answer: The series $\sum |a_n|$ converges. This means the series of absolute values also converges.

Flashcard 16: What does it mean if a series is absolutely convergent?

Answer: The series $\textstyle\bigsum |a_n|$ converges. This means the series of absolute values also converges.

Flashcard 17: State the limit condition for the Alternating Series Test.

Answer: $\textstyle\biglim_{n \to \text{inf}} a_n = 0$. The terms must approach zero for the series to have a chance at convergence.

Flashcard 18: Find if the series $\textstyle\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^5}$ converges.

Answer: Yes, it converges by the Alternating Series Test. Higher powers guarantee all alternating series test conditions are satisfied.

Flashcard 19: Find if $\textstyle\sum_{n=1}^{\infty} (-1)^n \frac{1}{\ln(n+1)}$ converges.

Answer: Yes, it converges by the Alternating Series Test. All conditions are met: positive terms, decreasing, limit to zero.

Flashcard 20: Find the error bound for $\textstyle \sum_{n=1}^{\infty} (-1)^n \frac{1}{n^2}$ at $N=3$.

Answer: Error $|R_3| < |a_4| = \frac{1}{16}$. The fourth term gives the error bound: $\frac{1}{4^2} = \frac{1}{16}$.

Flashcard 21: Does the series $\textstyle \sum_{n=1}^{\infty} (-1)^n (1 + \frac{1}{n^2})$ converge?

Answer: No, it does not converge. The limit $\lim_{n\to\infty} (1 + \frac{1}{n^2}) = 1 \neq 0$.

Flashcard 22: What is the significance of the term $(-1)^n$ in an alternating series?

Answer: It causes the series to alternate in sign. The $(-1)^n$ factor creates the required alternating pattern.

Flashcard 23: What happens if $a_n$ does not decrease to 0 in an alternating series?

Answer: The series diverges. The series fails to meet the necessary convergence conditions.

Flashcard 24: Find the limit of $a_n$ for $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{n}$.

Answer: Limit $a_n = \frac{1}{n} \to 0$ as $n \to \text{inf}$. As $n$ increases, $\frac{1}{n}$ approaches zero.

Flashcard 25: Find if $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^4}$ converges.

Answer: Yes, it converges by the Alternating Series Test. Even higher powers guarantee faster convergence to zero.

Flashcard 26: State the Alternating Series Remainder Theorem.

Answer: The remainder is less than the first unused term. This gives an upper bound on the approximation error.

Flashcard 27: What is the error bound for an alternating series?

Answer: Error $|R_N| < |a_{N+1}|$ for partial sum $S_N$. The error magnitude is bounded by the next term's absolute value.

Flashcard 28: Does the series $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{n}$ satisfy $a_n \to 0$?

Answer: Yes, $a_n = \frac{1}{n} \to 0$ as $n \to \text{inf}$. The harmonic terms $\frac{1}{n}$ clearly approach zero as $n \to \infty$.

Flashcard 29: Identify the condition required for terms in the Alternating Series.

Answer: The terms $a_n$ must be positive: $a_n > 0$. This ensures the series doesn't have negative terms interfering with convergence.

Flashcard 30: Does the series $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^2}$ converge?

Answer: Yes, it converges by the Alternating Series Test. All three conditions are satisfied: positive, decreasing, limit to zero.

Flashcard 31: What is the Alternating Series Estimation Theorem?

Answer: Provides error bound for partial sums of convergent series. It quantifies how close partial sums are to the series sum.

Flashcard 32: What does it mean for the terms to be 'eventually decreasing'?

Answer: There exists $N$ such that $a_{n+1} < a_n$ for $n > N$. The decreasing condition only needs to hold for sufficiently large $n$.

Flashcard 33: Determine if $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n+1}$ satisfies $a_n \to 0$.

Answer: Yes, $a_n = \frac{1}{n+1} \to 0$ as $n \to \infty$. The shifted harmonic terms still approach zero as required.

Flashcard 34: Determine if the series $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^3}$ converges.

Answer: Yes, it converges by the Alternating Series Test. Higher powers ensure faster convergence with all conditions satisfied.

Flashcard 35: Determine if the series $\textstyle \sum_{n=1}^{\infty} (-1)^n \frac{1}{n^3}$ converges.

Answer: Yes, it converges by the Alternating Series Test. Higher powers ensure faster convergence with all conditions satisfied.

Flashcard 36: What type of series does the Alternating Series Test apply to?

Answer: Applies to series with terms $(-1)^n a_n$. The alternating factor $(-1)^n$ creates the sign pattern.

Flashcard 37: Identify the condition required for terms in the Alternating Series.

Answer: The terms $a_n$ must be positive: $a_n > 0$. This ensures the series doesn't have negative terms interfering with convergence.

Flashcard 38: What is the condition for the terms $a_n$ to be considered decreasing?

Answer: $a_{n+1} < a_n$ for all $n$. This ensures the terms form a monotonically decreasing sequence.

Flashcard 39: Identify if $\textstyle\sum_{n=1}^{\infty} (-1)^n (1 + \frac{1}{n})$ converges.

Answer: No, it does not satisfy $a_n \to 0$ as $n \to \infty$. The limit is $1$, not $0$, so the series diverges.

Flashcard 40: Find if the series $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$ satisfies decreasing terms.

Answer: Yes, $\frac{1}{n+1} < \frac{1}{n}$ for $n \to \infty$. Since $n+1 > n$, the reciprocals decrease monotonically.

Flashcard 41: What is the significance of the term $(-1)^n$ in an alternating series?

Answer: It causes the series to alternate in sign. The $(-1)^n$ factor creates the required alternating pattern.

Flashcard 42: Identify the series that converges: (A) $\textstyle\bigsum (-1)^n \frac{1}{n^2}$, (B) $\textstyle\bigsum (-1)^n n$.

Answer: (A) converges by the Alternating Series Test. Series (A) satisfies all conditions while (B) has terms that don't approach zero.

Flashcard 43: What is the result if $a_n \to 0$ is not satisfied?

Answer: The series diverges. Without the limit condition, the series cannot converge.

Flashcard 44: State the primary purpose of the Alternating Series Test.

Answer: To determine if an alternating series converges. It checks the three key conditions for alternating series convergence.

Flashcard 45: Identify the series that converges: (A) $\sum (-1)^n \frac{1}{n^2}$, (B) $\sum (-1)^n n$.

Answer: (A) converges by the Alternating Series Test. Series (A) satisfies all conditions while (B) has terms that don't approach zero.

Flashcard 46: Find if $\textstyle \sum_{n=1}^{\infty} (-1)^n \frac{1}{n^4}$ converges.

Answer: Yes, it converges by the Alternating Series Test. Even higher powers guarantee faster convergence to zero.

Flashcard 47: What does the Alternating Series Test determine about a series?

Answer: Determines if an alternating series converges. It's the specific test for alternating series convergence criteria.

Flashcard 48: Find the error bound for $\textstyle\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^2}$ at $N=3$.

Answer: Error $|R_3| < |a_4| = \frac{1}{16}$. The fourth term gives the error bound: $\frac{1}{4^2} = \frac{1}{16}$.

Flashcard 49: Find if the series $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$ satisfies decreasing terms.

Answer: Yes, $\frac{1}{n+1} < \frac{1}{n}$ for $n \to \infty$. Since $n+1 > n$, the reciprocals decrease monotonically.

Flashcard 50: What does the Alternating Series Test determine about a series?

Answer: Determines if an alternating series converges. It's the specific test for alternating series convergence criteria.

Flashcard 51: What is the error bound for an alternating series?

Answer: Error $|R_N| < |a_{N+1}|$ for partial sum $S_N$. The error magnitude is bounded by the next term's absolute value.

Flashcard 52: What is the result if $a_n \to 0$ is not satisfied?

Answer: The series diverges. Without the limit condition, the series cannot converge.

Flashcard 53: Identify if $\textstyle\sum_{n=1}^{\infty} (-1)^n (1 + \frac{1}{n})$ converges.

Answer: No, it does not satisfy $a_n \to 0$ as $n \to \infty$. The limit is $1$, not $0$, so the series diverges.

Flashcard 54: Find the limit of $a_n$ for $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$.

Answer: Limit $a_n = \frac{1}{n} \to 0$ as $n \to \infty$. As $n$ increases, $\frac{1}{n}$ approaches zero.

Flashcard 55: State the primary purpose of the Alternating Series Test.

Answer: To determine if an alternating series converges. It checks the three key conditions for alternating series convergence.

Flashcard 56: Find if the series $\sum_{n=1}^{\infty} (-1)^n \frac{n}{n+1}$ converges.

Answer: No, it does not converge. The limit $\lim_{n\to\infty} \frac{n}{n+1} = 1 \neq 0$.

Flashcard 57: Identify the series that does not converge: (A) $\textstyle\sum (-1)^n \frac{1}{n^3}$, (B) $\textstyle\sum (-1)^n n$.

Answer: (B) does not converge. Series (B) has unbounded terms while (A) satisfies all conditions.

Flashcard 58: Find the error bound for $\textstyle\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$ at $N=2$.

Answer: Error $|R_2| < |a_3| = \frac{1}{3}$. The third term in the harmonic series: $\frac{1}{3}$.

Flashcard 59: Find the error bound for $\textstyle \sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$ at $N=2$.

Answer: Error $|R_2| < |a_3| = \frac{1}{3}$. The third term in the harmonic series: $\frac{1}{3}$

Flashcard 60: What is the condition for the terms $a_n$ to be considered decreasing?

Answer: $a_{n+1} < a_n$ for all $n$. This ensures the terms form a monotonically decreasing sequence.

Flashcard 61: Find if the series $\textstyle\sum_{n=1}^{\infty} (-1)^n \frac{n}{n+1}$ converges.

Answer: No, it does not converge. The limit $\lim_{n\to\infty} \frac{n}{n+1} = 1 \neq 0$.