Alternating Series Test for Convergence - AP Calculus BC
Card 1 of 30
Identify if $\textstyle\sum_{n=1}^{\infty} (-1)^n (1 + \frac{1}{n})$ converges.
Identify if $\textstyle\sum_{n=1}^{\infty} (-1)^n (1 + \frac{1}{n})$ converges.
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No, it does not satisfy $a_n \to 0$ as $n \to \infty$. The limit is $1$, not $0$, so the series diverges.
No, it does not satisfy $a_n \to 0$ as $n \to \infty$. The limit is $1$, not $0$, so the series diverges.
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Does the series $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{n}$ satisfy $a_n \to 0$?
Does the series $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{n}$ satisfy $a_n \to 0$?
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Yes, $a_n = \frac{1}{n} \to 0$ as $n \to \text{inf}$. The harmonic terms $\frac{1}{n}$ clearly approach zero as $n \to \infty$.
Yes, $a_n = \frac{1}{n} \to 0$ as $n \to \text{inf}$. The harmonic terms $\frac{1}{n}$ clearly approach zero as $n \to \infty$.
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What does the Alternating Series Test determine about a series?
What does the Alternating Series Test determine about a series?
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Determines if an alternating series converges. It's the specific test for alternating series convergence criteria.
Determines if an alternating series converges. It's the specific test for alternating series convergence criteria.
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What is the condition for the terms $a_n$ to be considered decreasing?
What is the condition for the terms $a_n$ to be considered decreasing?
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$a_{n+1} < a_n$ for all $n$. This ensures the terms form a monotonically decreasing sequence.
$a_{n+1} < a_n$ for all $n$. This ensures the terms form a monotonically decreasing sequence.
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Find the limit of $a_n$ for $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{n}$.
Find the limit of $a_n$ for $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{n}$.
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Limit $a_n = \frac{1}{n} \to 0$ as $n \to \text{inf}$. As $n$ increases, $\frac{1}{n}$ approaches zero.
Limit $a_n = \frac{1}{n} \to 0$ as $n \to \text{inf}$. As $n$ increases, $\frac{1}{n}$ approaches zero.
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Determine if the series $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^3}$ converges.
Determine if the series $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^3}$ converges.
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Yes, it converges by the Alternating Series Test. Higher powers ensure faster convergence with all conditions satisfied.
Yes, it converges by the Alternating Series Test. Higher powers ensure faster convergence with all conditions satisfied.
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State the limit condition for the Alternating Series Test.
State the limit condition for the Alternating Series Test.
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$\textstyle\biglim_{n \to \text{inf}} a_n = 0$. The terms must approach zero for the series to have a chance at convergence.
$\textstyle\biglim_{n \to \text{inf}} a_n = 0$. The terms must approach zero for the series to have a chance at convergence.
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Find if $\textstyle\sum_{n=1}^{\infty} (-1)^n \frac{1}{\ln(n+1)}$ converges.
Find if $\textstyle\sum_{n=1}^{\infty} (-1)^n \frac{1}{\ln(n+1)}$ converges.
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Yes, it converges by the Alternating Series Test. All conditions are met: positive terms, decreasing, limit to zero.
Yes, it converges by the Alternating Series Test. All conditions are met: positive terms, decreasing, limit to zero.
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Does the series $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{n^2}$ converge?
Does the series $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{n^2}$ converge?
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Yes, it converges by the Alternating Series Test. All three conditions are satisfied: positive, decreasing, limit to zero.
Yes, it converges by the Alternating Series Test. All three conditions are satisfied: positive, decreasing, limit to zero.
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What is the Alternating Series Estimation Theorem?
What is the Alternating Series Estimation Theorem?
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Provides error bound for partial sums of convergent series. It quantifies how close partial sums are to the series sum.
Provides error bound for partial sums of convergent series. It quantifies how close partial sums are to the series sum.
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What does it mean for the terms to be 'eventually decreasing'?
What does it mean for the terms to be 'eventually decreasing'?
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There exists $N$ such that $a_{n+1} < a_n$ for $n > N$. The decreasing condition only needs to hold for sufficiently large $n$.
There exists $N$ such that $a_{n+1} < a_n$ for $n > N$. The decreasing condition only needs to hold for sufficiently large $n$.
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State the Alternating Series Remainder Theorem.
State the Alternating Series Remainder Theorem.
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The remainder is less than the first unused term. This gives an upper bound on the approximation error.
The remainder is less than the first unused term. This gives an upper bound on the approximation error.
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State the primary purpose of the Alternating Series Test.
State the primary purpose of the Alternating Series Test.
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To determine if an alternating series converges. It checks the three key conditions for alternating series convergence.
To determine if an alternating series converges. It checks the three key conditions for alternating series convergence.
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What happens if $a_n$ does not decrease to 0 in an alternating series?
What happens if $a_n$ does not decrease to 0 in an alternating series?
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The series diverges. The series fails to meet the necessary convergence conditions.
The series diverges. The series fails to meet the necessary convergence conditions.
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What is the result if $a_n \to 0$ is not satisfied?
What is the result if $a_n \to 0$ is not satisfied?
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The series diverges. Without the limit condition, the series cannot converge.
The series diverges. Without the limit condition, the series cannot converge.
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Identify the series that does not converge: (A) $\textstyle\sum (-1)^n \frac{1}{n^3}$, (B) $\textstyle\sum (-1)^n n$.
Identify the series that does not converge: (A) $\textstyle\sum (-1)^n \frac{1}{n^3}$, (B) $\textstyle\sum (-1)^n n$.
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(B) does not converge. Series (B) has unbounded terms while (A) satisfies all conditions.
(B) does not converge. Series (B) has unbounded terms while (A) satisfies all conditions.
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Determine whether $\textstyle \sum_{n=1}^{\infty} (-1)^n n$ converges.
Determine whether $\textstyle \sum_{n=1}^{\infty} (-1)^n n$ converges.
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No, the terms do not approach zero. The terms $a_n = n$ grow without bound, violating the limit condition.
No, the terms do not approach zero. The terms $a_n = n$ grow without bound, violating the limit condition.
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Identify the condition required for terms in the Alternating Series.
Identify the condition required for terms in the Alternating Series.
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The terms $a_n$ must be positive: $a_n > 0$. This ensures the series doesn't have negative terms interfering with convergence.
The terms $a_n$ must be positive: $a_n > 0$. This ensures the series doesn't have negative terms interfering with convergence.
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Find if $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^4}$ converges.
Find if $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^4}$ converges.
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Yes, it converges by the Alternating Series Test. Even higher powers guarantee faster convergence to zero.
Yes, it converges by the Alternating Series Test. Even higher powers guarantee faster convergence to zero.
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Does the series $\textstyle \sum_{n=1}^{\infty} (-1)^n (1 + \frac{1}{n^2})$ converge?
Does the series $\textstyle \sum_{n=1}^{\infty} (-1)^n (1 + \frac{1}{n^2})$ converge?
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No, it does not converge. The limit $\lim_{n\to\infty} (1 + \frac{1}{n^2}) = 1 \neq 0$.
No, it does not converge. The limit $\lim_{n\to\infty} (1 + \frac{1}{n^2}) = 1 \neq 0$.
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Identify the series that converges: (A) $\sum (-1)^n \frac{1}{n^2}$, (B) $\sum (-1)^n n$.
Identify the series that converges: (A) $\sum (-1)^n \frac{1}{n^2}$, (B) $\sum (-1)^n n$.
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(A) converges by the Alternating Series Test. Series (A) satisfies all conditions while (B) has terms that don't approach zero.
(A) converges by the Alternating Series Test. Series (A) satisfies all conditions while (B) has terms that don't approach zero.
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Find if the series $\sum_{n=1}^{\infty} (-1)^n \frac{n}{n+1}$ converges.
Find if the series $\sum_{n=1}^{\infty} (-1)^n \frac{n}{n+1}$ converges.
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No, it does not converge. The limit $\lim_{n\to\infty} \frac{n}{n+1} = 1 \neq 0$.
No, it does not converge. The limit $\lim_{n\to\infty} \frac{n}{n+1} = 1 \neq 0$.
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What is the error bound for an alternating series?
What is the error bound for an alternating series?
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Error $ |R_N| < |a_{N+1}| $ for partial sum $ S_N $. The error magnitude is bounded by the next term's absolute value.
Error $ |R_N| < |a_{N+1}| $ for partial sum $ S_N $. The error magnitude is bounded by the next term's absolute value.
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What type of series does the Alternating Series Test apply to?
What type of series does the Alternating Series Test apply to?
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Applies to series with terms $(-1)^n a_n$. The alternating factor $(-1)^n$ creates the sign pattern.
Applies to series with terms $(-1)^n a_n$. The alternating factor $(-1)^n$ creates the sign pattern.
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State the Alternating Series Remainder Theorem.
State the Alternating Series Remainder Theorem.
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The remainder is less than the first unused term. This gives an upper bound on the approximation error.
The remainder is less than the first unused term. This gives an upper bound on the approximation error.
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State the primary purpose of the Alternating Series Test.
State the primary purpose of the Alternating Series Test.
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To determine if an alternating series converges. It checks the three key conditions for alternating series convergence.
To determine if an alternating series converges. It checks the three key conditions for alternating series convergence.
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Find if the series $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$ satisfies decreasing terms.
Find if the series $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$ satisfies decreasing terms.
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Yes, $\frac{1}{n+1} < \frac{1}{n}$ for $n \to \infty$. Since $n+1 > n$, the reciprocals decrease monotonically.
Yes, $\frac{1}{n+1} < \frac{1}{n}$ for $n \to \infty$. Since $n+1 > n$, the reciprocals decrease monotonically.
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Does the series $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^2}$ converge?
Does the series $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n^2}$ converge?
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Yes, it converges by the Alternating Series Test. All three conditions are satisfied: positive, decreasing, limit to zero.
Yes, it converges by the Alternating Series Test. All three conditions are satisfied: positive, decreasing, limit to zero.
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Identify the condition required for terms in the Alternating Series.
Identify the condition required for terms in the Alternating Series.
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The terms $a_n$ must be positive: $a_n > 0$. This ensures the series doesn't have negative terms interfering with convergence.
The terms $a_n$ must be positive: $a_n > 0$. This ensures the series doesn't have negative terms interfering with convergence.
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Find if $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{\text{ln}(n+1)}$ converges.
Find if $\textstyle\bigsum_{n=1}^{\text{inf}} (-1)^n \frac{1}{\text{ln}(n+1)}$ converges.
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Yes, it converges by the Alternating Series Test. All conditions are met: positive terms, decreasing, limit to zero.
Yes, it converges by the Alternating Series Test. All conditions are met: positive terms, decreasing, limit to zero.
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