Evaluate the infinite series by identifying it as the value of an integral of a geometric series. (– 1)" 8"+1(n + 1) 2 In (3) x n=0 n+1 | f(t)dt where f(æ) = E(- 1)" Hint: Write it as 8 8 +x n=0
Evaluate the infinite series by identifying it as the value of an integral of a geometric series. (– 1)" 8"+1(n + 1) 2 In (3) x n=0 n+1 | f(t)dt where f(æ) = E(- 1)" Hint: Write it as 8 8 +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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![**Evaluate the Infinite Series by Identifying it as the Value of an Integral of a Geometric Series**
\[
\sum_{n=0}^{\infty} \frac{(-1)^n}{8^{n+1}(n+1)} = \boxed{2 \ln(3)} \times
\]
**Hint:** Write it as \(\int_0^1 f(t) \, dt\) where \(f(x) = \sum_{n=0}^{\infty} (-1)^n \left( \frac{1}{8} \right)^{n+1} x^n = \frac{1}{8 + x}\).
This equation involves transforming an infinite series into an integral representation. The goal is to express the series using an integral function \(f(x)\) that simplifies to a known series sum formula. The geometric series is utilized in the expression \(f(x) = \frac{1}{8 + x}\), which reflects the standard formula for an infinite geometric series sum. By solving the integral, you identify the value of the infinite series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb555586-9f31-4c15-934a-3aec181a2382%2Ff6e33d1e-6705-4d3e-b9dd-0514ea553bb0%2Foct7juk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluate the Infinite Series by Identifying it as the Value of an Integral of a Geometric Series**
\[
\sum_{n=0}^{\infty} \frac{(-1)^n}{8^{n+1}(n+1)} = \boxed{2 \ln(3)} \times
\]
**Hint:** Write it as \(\int_0^1 f(t) \, dt\) where \(f(x) = \sum_{n=0}^{\infty} (-1)^n \left( \frac{1}{8} \right)^{n+1} x^n = \frac{1}{8 + x}\).
This equation involves transforming an infinite series into an integral representation. The goal is to express the series using an integral function \(f(x)\) that simplifies to a known series sum formula. The geometric series is utilized in the expression \(f(x) = \frac{1}{8 + x}\), which reflects the standard formula for an infinite geometric series sum. By solving the integral, you identify the value of the infinite series.
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