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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
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Find the sum of each series.
![The image contains a mathematical expression and a series.
**Expression:**
\[
\left( \frac{1}{2} + \frac{1}{4} \right) + \left( \frac{1}{2^2} + \frac{1}{4^2} \right) + \left( \frac{1}{2^3} + \frac{1}{4^3} \right) + \ldots + \left( \frac{1}{2^k} + \frac{1}{4^k} \right) + \ldots
\]
This expression displays the sum of several terms, each consisting of a pair of fractions. The denominators in these fractions are powers of 2 and 4, respectively. The pattern continues indefinitely.
**Series:**
\[
\sum_{k=1}^{\infty} \left( \frac{1}{5^k} - \frac{1}{k(k+1)} \right)
\]
This series represents an infinite sum with each term given by the difference between \(\frac{1}{5^k}\) and \(\frac{1}{k(k+1)}\). The series starts with \(k = 1\) and continues to infinity. It combines geometric progression with another form involving an arithmetic sequence in the denominator.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80f7a2c1-bbd9-492a-a9a7-34302e2af2a3%2F4be0cfb0-b376-4057-8626-761614b2b29b%2Fxy1k9k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical expression and a series.
**Expression:**
\[
\left( \frac{1}{2} + \frac{1}{4} \right) + \left( \frac{1}{2^2} + \frac{1}{4^2} \right) + \left( \frac{1}{2^3} + \frac{1}{4^3} \right) + \ldots + \left( \frac{1}{2^k} + \frac{1}{4^k} \right) + \ldots
\]
This expression displays the sum of several terms, each consisting of a pair of fractions. The denominators in these fractions are powers of 2 and 4, respectively. The pattern continues indefinitely.
**Series:**
\[
\sum_{k=1}^{\infty} \left( \frac{1}{5^k} - \frac{1}{k(k+1)} \right)
\]
This series represents an infinite sum with each term given by the difference between \(\frac{1}{5^k}\) and \(\frac{1}{k(k+1)}\). The series starts with \(k = 1\) and continues to infinity. It combines geometric progression with another form involving an arithmetic sequence in the denominator.
Expert Solution
Step 1
First we find nature of given series.
Then we find its sum.
Step by step
Solved in 3 steps with 2 images
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