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Find the sum of the convergent series 00 n Σ (4) Σ

Find the sum of the convergent series 00 n Σ (4) Σ

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 Statement Find the sum of the convergent series: \[ \sum_{n=0}^{\infty} 6 \left( \frac{6}{7} \right)^n \] ### Explanation This is an infinite geometric series. The series can be expressed in the form: \[ \sum_{n=0}^{\infty} ar^n \] where \( a = 6 \) is the first term and \( r = \frac{6}{7} \) is the common ratio. For a geometric series to converge, the absolute value of the common ratio \( |r| \) should be less than 1. In this case, \( |r| = \frac{6}{7} < 1 \), so the series converges. #### Sum of the Series The sum \( S \) of an infinite geometric series is given by the formula: \[ S = \frac{a}{1 - r} \] Substituting the values: \[ S = \frac{6}{1 - \frac{6}{7}} = \frac{6}{\frac{1}{7}} = 6 \times 7 = 42 \] Therefore, the sum of the series is 42.
Transcribed Image Text:## Problem Statement Find the sum of the convergent series: \[ \sum_{n=0}^{\infty} 6 \left( \frac{6}{7} \right)^n \] ### Explanation This is an infinite geometric series. The series can be expressed in the form: \[ \sum_{n=0}^{\infty} ar^n \] where \( a = 6 \) is the first term and \( r = \frac{6}{7} \) is the common ratio. For a geometric series to converge, the absolute value of the common ratio \( |r| \) should be less than 1. In this case, \( |r| = \frac{6}{7} < 1 \), so the series converges. #### Sum of the Series The sum \( S \) of an infinite geometric series is given by the formula: \[ S = \frac{a}{1 - r} \] Substituting the values: \[ S = \frac{6}{1 - \frac{6}{7}} = \frac{6}{\frac{1}{7}} = 6 \times 7 = 42 \] Therefore, the sum of the series is 42.
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\sum _{n=0}^{\infty }\:6\left(\frac{6}{7}\right)^n

=6\cdot \sum _{n=0}^{\infty \:}\left(\frac{6}{7}\right)^n

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