Applying the ratio test to the series you would compute Hence the series diverges lim k 100 ak+1 ak lim k 100 f. 5k-1 k1(k+ 1)².4k¹
Applying the ratio test to the series you would compute Hence the series diverges lim k 100 ak+1 ak lim k 100 f. 5k-1 k1(k+ 1)².4k¹
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can you show me how to solve this? I am mostly getting stuck on how to convert this problem into an+1 I think
![### Applying the Ratio Test to the Series
Consider the series:
\[
\sum_{k=1}^{\infty} \frac{5^{k-1}}{(k+1)^2 \cdot 4^k}
\]
To determine the convergence or divergence of this series, you would compute:
\[
\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty}
\]
[Here, an expression for the limit should be calculated, and the result will be entered in the blank spaces provided.]
The determination of the series' behavior is as follows:
Hence, the series **diverges**.
### Explanation
The expression involves a summation of terms based on the function \(\frac{5^{k-1}}{(k+1)^2 \cdot 4^k}\). By applying the ratio test, evaluate the limit of the absolute value of the ratio between consecutive terms as \(k\) approaches infinity. If the result of the limit is greater than 1, the series diverges.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F65e9754d-e7bf-4951-a913-18b478a6d2d4%2F9142cfbc-5839-427f-9966-a4ce16d5565c%2Ffmcw3uk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Applying the Ratio Test to the Series
Consider the series:
\[
\sum_{k=1}^{\infty} \frac{5^{k-1}}{(k+1)^2 \cdot 4^k}
\]
To determine the convergence or divergence of this series, you would compute:
\[
\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty}
\]
[Here, an expression for the limit should be calculated, and the result will be entered in the blank spaces provided.]
The determination of the series' behavior is as follows:
Hence, the series **diverges**.
### Explanation
The expression involves a summation of terms based on the function \(\frac{5^{k-1}}{(k+1)^2 \cdot 4^k}\). By applying the ratio test, evaluate the limit of the absolute value of the ratio between consecutive terms as \(k\) approaches infinity. If the result of the limit is greater than 1, the series diverges.
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