By the alternating series test, the series First find the partial fraction decomposition of (六)( +10 k 4(−1)²+1 k=1k(k+ 10) 4 k(k+ 10) 4 k(k + 10) Then find the limit of the partial sums. 4(−1)k+1 k(k + 10) k=1 Enter your answer for the sum as a reduced fraction. converges. Find its sum.
By the alternating series test, the series First find the partial fraction decomposition of (六)( +10 k 4(−1)²+1 k=1k(k+ 10) 4 k(k+ 10) 4 k(k + 10) Then find the limit of the partial sums. 4(−1)k+1 k(k + 10) k=1 Enter your answer for the sum as a reduced fraction. converges. Find its sum.
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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Question
![By the alternating series test, the series
\[
\sum_{k=1}^{\infty} \frac{4(-1)^{k+1}}{k(k+10)}
\]
converges. Find its sum.
**First, find the partial fraction decomposition of**
\[
\frac{4}{k(k+10)}.
\]
\[
\frac{4}{k(k+10)} = \left( \frac{4}{10} \right) \left( \frac{1}{k} - \frac{1}{k+10} \right)
\]
**Then find the limit of the partial sums.**
\[
\sum_{k=1}^{\infty} \frac{4(-1)^{k+1}}{k(k+10)} = \boxed{\phantom{insert answer here}}
\]
Enter your answer for the sum as a reduced fraction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8109dbb5-39b2-442d-b17d-793e299fca4a%2F02e05633-af81-4b89-b58f-1d79e52c5e23%2Ff3olzmp_processed.png&w=3840&q=75)
Transcribed Image Text:By the alternating series test, the series
\[
\sum_{k=1}^{\infty} \frac{4(-1)^{k+1}}{k(k+10)}
\]
converges. Find its sum.
**First, find the partial fraction decomposition of**
\[
\frac{4}{k(k+10)}.
\]
\[
\frac{4}{k(k+10)} = \left( \frac{4}{10} \right) \left( \frac{1}{k} - \frac{1}{k+10} \right)
\]
**Then find the limit of the partial sums.**
\[
\sum_{k=1}^{\infty} \frac{4(-1)^{k+1}}{k(k+10)} = \boxed{\phantom{insert answer here}}
\]
Enter your answer for the sum as a reduced fraction.
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