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Problem 6. Find the sum of the seriesx", where x < 1. n=1

Problem 6. Find the sum of the seriesx", where x < 1. n=1

Calculus: Early Transcendentals
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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**Problem 6.** Find the sum of the series \(\sum_{n=1}^{\infty} x^n\), where \(|x| < 1\). *Explanation:* This problem involves an infinite geometric series with a common ratio \(x\), where the absolute value of \(x\) is less than 1. The general formula for the sum of an infinite geometric series \(a + ar + ar^2 + \ldots\) with \(|r| < 1\) is \(\frac{a}{1 - r}\), where \(a\) is the first term. Here, the series starts at \(n=1\), so the first term \(a = x\) and the common ratio \(r = x\). Thus, the sum of the series can be calculated as \(\frac{x}{1 - x}\).
Transcribed Image Text:**Problem 6.** Find the sum of the series \(\sum_{n=1}^{\infty} x^n\), where \(|x| < 1\). *Explanation:* This problem involves an infinite geometric series with a common ratio \(x\), where the absolute value of \(x\) is less than 1. The general formula for the sum of an infinite geometric series \(a + ar + ar^2 + \ldots\) with \(|r| < 1\) is \(\frac{a}{1 - r}\), where \(a\) is the first term. Here, the series starts at \(n=1\), so the first term \(a = x\) and the common ratio \(r = x\). Thus, the sum of the series can be calculated as \(\frac{x}{1 - x}\).
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