Evaluate the infinite series by identifying it as the value of an integral of a geometric series. (- 1)" 57+ (n + 1) 00 n=0 n+1 1 Hint: Write it as | f(t)dt where f(x) : %3D %3D 5+ x n=0
Evaluate the infinite series by identifying it as the value of an integral of a geometric series. (- 1)" 57+ (n + 1) 00 n=0 n+1 1 Hint: Write it as | f(t)dt where f(x) : %3D %3D 5+ x n=0
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Evaluating the Infinite Series Using Integration**
To find the value of the series, express it in terms of an integral related to a geometric series.
### Infinite Series
\[
\sum_{n=0}^{\infty} \frac{(-1)^n}{5^{n+1}(n+1)} = \, ?
\]
### Hint: Approach the Problem
Rewrite the series as an integral:
\[
\int_0^1 f(t) \, dt \quad \text{where} \quad f(x) = \sum_{n=0}^{\infty} (-1)^n \left( \frac{1}{5} \right)^{n+1} x^n = \frac{1}{5 + x}.
\]
### Explanation
- **Series Representation:**
- The given series involves terms of the form \((-1)^n \left(\frac{1}{5}\right)^{n+1} x^n\). This resembles a geometric series in \(x\).
- **Function Representation:**
- The function \(f(x) = \frac{1}{5 + x}\) is derived from recognizing that the series of \((-1)^n \left(\frac{1}{5}\right)^{n+1} x^n\) evaluates to a well-known closed form using geometric series sum formula for \(|x| < 1\).
- **Integration from 0 to 1:**
- The integral of \(f(t)\) from 0 to 1 will help in evaluating the infinite series through direct integration techniques, potentially transforming the summation into an integral in terms of a variable \(t\).
This combined knowledge allows the infinite series to be converted and evaluated effectively by linking integration techniques with geometric series properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc6eea0f-cfa5-4eda-a078-cdf97fe23765%2Fd18e60ee-96d3-4a11-a6fb-a34b2361065f%2Faq67as_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluating the Infinite Series Using Integration**
To find the value of the series, express it in terms of an integral related to a geometric series.
### Infinite Series
\[
\sum_{n=0}^{\infty} \frac{(-1)^n}{5^{n+1}(n+1)} = \, ?
\]
### Hint: Approach the Problem
Rewrite the series as an integral:
\[
\int_0^1 f(t) \, dt \quad \text{where} \quad f(x) = \sum_{n=0}^{\infty} (-1)^n \left( \frac{1}{5} \right)^{n+1} x^n = \frac{1}{5 + x}.
\]
### Explanation
- **Series Representation:**
- The given series involves terms of the form \((-1)^n \left(\frac{1}{5}\right)^{n+1} x^n\). This resembles a geometric series in \(x\).
- **Function Representation:**
- The function \(f(x) = \frac{1}{5 + x}\) is derived from recognizing that the series of \((-1)^n \left(\frac{1}{5}\right)^{n+1} x^n\) evaluates to a well-known closed form using geometric series sum formula for \(|x| < 1\).
- **Integration from 0 to 1:**
- The integral of \(f(t)\) from 0 to 1 will help in evaluating the infinite series through direct integration techniques, potentially transforming the summation into an integral in terms of a variable \(t\).
This combined knowledge allows the infinite series to be converted and evaluated effectively by linking integration techniques with geometric series properties.
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