Use the Ratio Test to determine whether [tex]$\sum_{n=1}^{\infty} a_n$[/tex] converges, where [tex]$a_n$[/tex] is given. State if the ratio test is inconclusive.
[tex]$\sum_{n=1}^{\infty} \frac{(2 n)!}{\left(\frac{n}{e}\right)^{2 n}}$[/tex]
Identify [tex]$a_n$[/tex].
[tex]$\frac{(2 n)!}{\left(\frac{n}{e}\right)^{2 n}}$[/tex]
Evaluate the following limit.
[tex]$\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_n}\right|$[/tex]
Since [tex]$\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_n}\right|$[/tex] ? Select.
Answer (1)
Compute a n + 1 based on the given a n .
Calculate the ratio a n a n + 1 .
Evaluate the limit lim n → ∞ a n a n + 1 , which equals 4.
Since the limit is greater than 1, the series diverges by the Ratio Test. 4
Explanation
Problem Setup We are given the series ∑ n = 1 ∞ a n where a n = ( e n ) 2 n ( 2 n )! . We want to use the Ratio Test to determine if the series converges, diverges, or if the test is inconclusive. The Ratio Test involves evaluating the limit lim n → ∞ a n a n + 1 .
Finding a_{n+1} First, we need to find a n + 1 . We have a n + 1 = ( e n + 1 ) 2 ( n + 1 ) ( 2 ( n + 1 ))! = ( e n + 1 ) 2 n + 2 ( 2 n + 2 )!
Computing the Ratio Next, we compute the ratio a n a n + 1 :
a n a n + 1 = ( e n ) 2 n ( 2 n )! ( e n + 1 ) 2 n + 2 ( 2 n + 2 )! = ( 2 n )! ( e n + 1 ) 2 n + 2 ( 2 n + 2 )! ( e n ) 2 n = ( 2 n )! ( e n + 1 ) 2 n ( e n + 1 ) 2 ( 2 n + 2 ) ( 2 n + 1 ) ( 2 n )! ( e n ) 2 n Simplifying, we get a n a n + 1 = ( n n + 1 ) 2 n ( e n + 1 ) 2 ( 2 n + 2 ) ( 2 n + 1 ) = ( 1 + n 1 ) 2 n e 2 ( n + 1 ) 2 4 n 2 + 6 n + 2 = ( 1 + n 1 ) 2 n ( n 2 + 2 n + 1 ) e 2 ( 4 n 2 + 6 n + 2 )
Evaluating the Limit Now, we evaluate the limit as n → ∞ :
n → ∞ lim a n a n + 1 = n → ∞ lim ( 1 + n 1 ) 2 n ( n 2 + 2 n + 1 ) e 2 ( 4 n 2 + 6 n + 2 ) We know that lim n → ∞ ( 1 + n 1 ) n = e , so lim n → ∞ ( 1 + n 1 ) 2 n = e 2 . Therefore, n → ∞ lim a n a n + 1 = n → ∞ lim e 2 ( n 2 + 2 n + 1 ) e 2 ( 4 n 2 + 6 n + 2 ) = n → ∞ lim n 2 + 2 n + 1 4 n 2 + 6 n + 2 = 4
Conclusion Since the limit is 4, which is greater than 1, the series diverges by the Ratio Test.
Examples
The Ratio Test is a powerful tool used to determine the convergence or divergence of infinite series. In fields like physics and engineering, infinite series are often used to model complex systems. For example, when analyzing the stability of a structure under varying loads, engineers might use the Ratio Test to ensure that a series representing the system's response converges, indicating stability. Similarly, in signal processing, the convergence of Fourier series, which decompose signals into simpler components, can be assessed using the Ratio Test to guarantee accurate signal reconstruction.
