Ratio Test for Convergence - AP Calculus BC
Card 1 of 30
What does $L = 1$ imply about the convergence of the series?
What does $L = 1$ imply about the convergence of the series?
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The test is inconclusive. The boundary case provides no information about convergence.
The test is inconclusive. The boundary case provides no information about convergence.
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Determine the Ratio Test result for $a_n = n^3 e^{-n}$.
Determine the Ratio Test result for $a_n = n^3 e^{-n}$.
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Convergent. $L = \frac{(n+1)^3}{n^3} \cdot \frac{1}{e} = \frac{1}{e} < 1$.
Convergent. $L = \frac{(n+1)^3}{n^3} \cdot \frac{1}{e} = \frac{1}{e} < 1$.
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Does $a_n = \frac{n^2}{(n+1)!}$ converge according to the Ratio Test?
Does $a_n = \frac{n^2}{(n+1)!}$ converge according to the Ratio Test?
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Convergent. $L = \frac{(n+1)^2}{(n+2)!} \cdot \frac{(n+1)!}{n^2} = \frac{(n+1)^2}{n^2(n+2)} \to 0 < 1$.
Convergent. $L = \frac{(n+1)^2}{(n+2)!} \cdot \frac{(n+1)!}{n^2} = \frac{(n+1)^2}{n^2(n+2)} \to 0 < 1$.
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Evaluate convergence of $a_n = \frac{2^n}{n!}$ using the Ratio Test.
Evaluate convergence of $a_n = \frac{2^n}{n!}$ using the Ratio Test.
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Convergent. $L = \frac{2}{n+1} \to 0 < 1$.
Convergent. $L = \frac{2}{n+1} \to 0 < 1$.
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Determine the convergence of $a_n = 3^n \cdot n!$ using the Ratio Test.
Determine the convergence of $a_n = 3^n \cdot n!$ using the Ratio Test.
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Divergent. $L = 3(n+1) \to \infty > 1$.
Divergent. $L = 3(n+1) \to \infty > 1$.
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Does $a_n = \frac{n}{2^n}$ converge according to the Ratio Test?
Does $a_n = \frac{n}{2^n}$ converge according to the Ratio Test?
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Convergent. $L = \frac{n+1}{2n} = \frac{1}{2} < 1$.
Convergent. $L = \frac{n+1}{2n} = \frac{1}{2} < 1$.
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Does $a_n = \frac{n^2}{3^n}$ converge according to the Ratio Test?
Does $a_n = \frac{n^2}{3^n}$ converge according to the Ratio Test?
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Convergent. $L = \frac{(n+1)^2}{3n^2} = \frac{1}{3} < 1$.
Convergent. $L = \frac{(n+1)^2}{3n^2} = \frac{1}{3} < 1$.
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Identify the result of the Ratio Test for $a_n = \frac{3^n}{n!}$.
Identify the result of the Ratio Test for $a_n = \frac{3^n}{n!}$.
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Convergent. $L = \frac{3}{n+1} \to 0 < 1$.
Convergent. $L = \frac{3}{n+1} \to 0 < 1$.
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What is the convergence outcome for $a_n = \frac{n}{3^n}$ using the Ratio Test?
What is the convergence outcome for $a_n = \frac{n}{3^n}$ using the Ratio Test?
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Convergent. $L = \frac{n+1}{3n} = \frac{1}{3} < 1$.
Convergent. $L = \frac{n+1}{3n} = \frac{1}{3} < 1$.
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Evaluate $a_n = \frac{n!}{2^n}$ using the Ratio Test. Convergent or divergent?
Evaluate $a_n = \frac{n!}{2^n}$ using the Ratio Test. Convergent or divergent?
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Divergent. $L = \frac{n+1}{2} \to \infty > 1$.
Divergent. $L = \frac{n+1}{2} \to \infty > 1$.
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What does the Ratio Test say about $a_n = \frac{1}{(n+1)!}$?
What does the Ratio Test say about $a_n = \frac{1}{(n+1)!}$?
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Convergent. $L = \frac{1}{n+2} \to 0 < 1$.
Convergent. $L = \frac{1}{n+2} \to 0 < 1$.
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State the Ratio Test conclusion for $a_n = \frac{2^n}{n!}$.
State the Ratio Test conclusion for $a_n = \frac{2^n}{n!}$.
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Convergent. $L = \frac{2}{n+1} \to 0 < 1$.
Convergent. $L = \frac{2}{n+1} \to 0 < 1$.
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Evaluate $a_n = \frac{n^3}{n!}$ using the Ratio Test. Convergent or divergent?
Evaluate $a_n = \frac{n^3}{n!}$ using the Ratio Test. Convergent or divergent?
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Convergent. $L = \frac{(n+1)^3}{n^3} \cdot \frac{1}{n+1} = \frac{(n+1)^2}{n^3} \to 0 < 1$.
Convergent. $L = \frac{(n+1)^3}{n^3} \cdot \frac{1}{n+1} = \frac{(n+1)^2}{n^3} \to 0 < 1$.
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Identify the convergence of $a_n = \frac{1}{3^n}$ using the Ratio Test.
Identify the convergence of $a_n = \frac{1}{3^n}$ using the Ratio Test.
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Convergent. $L = \frac{1}{3} < 1$.
Convergent. $L = \frac{1}{3} < 1$.
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What does the Ratio Test conclude for $a_n = n^2 e^{-n}$?
What does the Ratio Test conclude for $a_n = n^2 e^{-n}$?
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Convergent. $L = \frac{(n+1)^2}{n^2} \cdot \frac{1}{e} = \frac{1}{e} < 1$.
Convergent. $L = \frac{(n+1)^2}{n^2} \cdot \frac{1}{e} = \frac{1}{e} < 1$.
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What is the result of Ratio Test for $a_n = \frac{n!}{n^n}$?
What is the result of Ratio Test for $a_n = \frac{n!}{n^n}$?
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Convergent. $L = \frac{n+1}{n} \cdot \frac{n^n}{(n+1)^{n+1}} = \frac{1}{e} < 1$.
Convergent. $L = \frac{n+1}{n} \cdot \frac{n^n}{(n+1)^{n+1}} = \frac{1}{e} < 1$.
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State the Ratio Test result for $a_n = \frac{(n+1)!}{n!}$.
State the Ratio Test result for $a_n = \frac{(n+1)!}{n!}$.
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Divergent. $L = \frac{(n+2)!}{(n+1)!} = n+2 \to \infty > 1$.
Divergent. $L = \frac{(n+2)!}{(n+1)!} = n+2 \to \infty > 1$.
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What is the Ratio Test outcome for $a_n = n!$?
What is the Ratio Test outcome for $a_n = n!$?
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Divergent. $L = n+1 \to \infty > 1$.
Divergent. $L = n+1 \to \infty > 1$.
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Is $a_n = \frac{5^n}{n!}$ convergent by the Ratio Test?
Is $a_n = \frac{5^n}{n!}$ convergent by the Ratio Test?
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Convergent. $L = \frac{5}{n+1} \to 0 < 1$.
Convergent. $L = \frac{5}{n+1} \to 0 < 1$.
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What condition must $L$ satisfy for the series to diverge?
What condition must $L$ satisfy for the series to diverge?
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$L > 1$ or $L = \infty$. When the ratio exceeds 1, terms grow without bound.
$L > 1$ or $L = \infty$. When the ratio exceeds 1, terms grow without bound.
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What condition must $L$ satisfy for the series to converge?
What condition must $L$ satisfy for the series to converge?
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$L < 1$. When the ratio is less than 1, terms shrink fast enough.
$L < 1$. When the ratio is less than 1, terms shrink fast enough.
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State the formula used in the Ratio Test for a series $a_n$.
State the formula used in the Ratio Test for a series $a_n$.
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$L = \frac{|a_{n+1}|}{|a_n|}$ as $n \to \infty$. This is the limit of consecutive term ratios.
$L = \frac{|a_{n+1}|}{|a_n|}$ as $n \to \infty$. This is the limit of consecutive term ratios.
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Evaluate convergence of $a_n = \frac{2^n}{n!}$ using the Ratio Test.
Evaluate convergence of $a_n = \frac{2^n}{n!}$ using the Ratio Test.
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Convergent. $L = \frac{2}{n+1} \to 0 < 1$.
Convergent. $L = \frac{2}{n+1} \to 0 < 1$.
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Is $a_n = \frac{n!}{n^n}$ convergent by the Ratio Test?
Is $a_n = \frac{n!}{n^n}$ convergent by the Ratio Test?
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Convergent. $L = \frac{n+1}{n} \cdot \frac{n^n}{(n+1)^{n+1}} = \frac{1}{e} < 1$.
Convergent. $L = \frac{n+1}{n} \cdot \frac{n^n}{(n+1)^{n+1}} = \frac{1}{e} < 1$.
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Apply the Ratio Test: $a_n = \frac{n!}{3^n}$. Convergent or divergent?
Apply the Ratio Test: $a_n = \frac{n!}{3^n}$. Convergent or divergent?
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Divergent. $L = \frac{n+1}{3} \to \infty > 1$.
Divergent. $L = \frac{n+1}{3} \to \infty > 1$.
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Apply the Ratio Test: $a_n = \frac{n^2}{n!}$. Convergent or divergent?
Apply the Ratio Test: $a_n = \frac{n^2}{n!}$. Convergent or divergent?
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Convergent. $L = \frac{(n+1)^2}{n^2} \cdot \frac{1}{n+1} = \frac{n+1}{n^2} \to 0 < 1$.
Convergent. $L = \frac{(n+1)^2}{n^2} \cdot \frac{1}{n+1} = \frac{n+1}{n^2} \to 0 < 1$.
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Apply the Ratio Test: $a_n = \frac{5^n}{n^2}$. Convergent or divergent?
Apply the Ratio Test: $a_n = \frac{5^n}{n^2}$. Convergent or divergent?
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Divergent. $L = 5 \cdot \frac{n^2}{(n+1)^2} = 5 > 1$.
Divergent. $L = 5 \cdot \frac{n^2}{(n+1)^2} = 5 > 1$.
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State the limit test result for $a_n = \frac{n^2}{2^n}$ using the Ratio Test.
State the limit test result for $a_n = \frac{n^2}{2^n}$ using the Ratio Test.
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Convergent. $L = \frac{(n+1)^2}{n^2} \cdot \frac{1}{2} = \frac{1}{2} < 1$.
Convergent. $L = \frac{(n+1)^2}{n^2} \cdot \frac{1}{2} = \frac{1}{2} < 1$.
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Identify the series type: $a_n = \frac{1}{n!}$. Use the Ratio Test.
Identify the series type: $a_n = \frac{1}{n!}$. Use the Ratio Test.
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Convergent. $L = \frac{1}{(n+1)} \to 0 < 1$.
Convergent. $L = \frac{1}{(n+1)} \to 0 < 1$.
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State the Ratio Test result for $a_n = \frac{(n+1)!}{n!}$.
State the Ratio Test result for $a_n = \frac{(n+1)!}{n!}$.
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Divergent. $L = \frac{(n+2)!}{(n+1)!} = n+2 \to \infty > 1$.
Divergent. $L = \frac{(n+2)!}{(n+1)!} = n+2 \to \infty > 1$.
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