Use either the ratio test or the root test as appropriate to determine whether the series [tex]$\sum_{n=1}^{\infty} a_n$[/tex] with given terms [tex]$a_n$[/tex] converges, or state if the test is inconclusive.
[tex]$a_n=\frac{n!}{1 \cdot 3 \cdot 5 \cdots(2 n-1)}$[/tex]
A. The series converges.
B. The series diverges.
C. The test is inconclusive.
Answer (2)
Apply the ratio test to the series ∑ n = 1 ∞ a n with a n = 1 ⋅ 3 ⋅ 5 ⋯ ( 2 n − 1 ) n ! .
Calculate the ratio a n a n + 1 = 2 n + 1 n + 1 .
Find the limit as n approaches infinity: lim n → ∞ 2 n + 1 n + 1 = 2 1 .
Since the limit is 2 1 < 1 , the series converges by the ratio test. The series converges.
Explanation
Problem Analysis We are given the series ∑ n = 1 ∞ a n with a n = 1 ⋅ 3 ⋅ 5 ⋯ ( 2 n − 1 ) n ! . Our goal is to determine whether this series converges, diverges, or if the test is inconclusive, using either the ratio test or the root test.
Applying the Ratio Test Let's apply the ratio test. The ratio test states that if lim n → ∞ ∣ a n a n + 1 ∣ < 1 , the series converges; if the limit is 1"> > 1 , the series diverges; and if the limit is equal to 1, the test is inconclusive.
Calculating the Ratio We need to find the ratio a n a n + 1 . We have:
a n a n + 1 = 1 ⋅ 3 ⋅ 5 ⋯ ( 2 n − 1 ) n ! 1 ⋅ 3 ⋅ 5 ⋯ ( 2 ( n + 1 ) − 1 ) ( n + 1 )!
Simplifying this expression, we get:
a n a n + 1 = n ! ( n + 1 )! ⋅ 1 ⋅ 3 ⋅ 5 ⋯ ( 2 n − 1 ) ( 2 ( n + 1 ) − 1 ) 1 ⋅ 3 ⋅ 5 ⋯ ( 2 n − 1 ) = ( n + 1 ) ⋅ 2 n + 1 1 = 2 n + 1 n + 1
Evaluating the Limit Now, we need to find the limit of this ratio as n approaches infinity:
n → ∞ lim a n a n + 1 = n → ∞ lim 2 n + 1 n + 1
To evaluate this limit, we can divide both the numerator and the denominator by n :
n → ∞ lim 2 + n 1 1 + n 1 = 2 1
Conclusion Since lim n → ∞ ∣ a n a n + 1 ∣ = 2 1 < 1 , the series converges by the ratio test.
Examples
Consider a scenario where you are analyzing the efficiency of an algorithm. If the number of steps the algorithm takes can be represented by a series, determining whether that series converges can tell you if the algorithm will complete in a finite amount of time or not. In this case, the convergence of the series indicates the algorithm's efficiency and its ability to produce a result in a reasonable time frame.
Using the ratio test for the series ∑ n = 1 ∞ a n where a n = 1 ⋅ 3 ⋅ 5 ⋯ ( 2 n − 1 ) n ! , we found that L = 2 1 < 1 , which means the series converges. Thus, the correct option is A. The series converges.
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