Question

Evaluate $\sum_{k=0}^{\infty} r(1-r)^{k}, 0<r<2.$

   Evaluate $\sum_{k=0}^{\infty} r(1-r)^{k}, 0<r<2.$

Calculus

Dale Varberg, Edwin… 9th Edition

Chapter 9, Problem 21

Instant Answer

Solved by Expert Vipender Yadav

06/19/2021

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Step 1: The given series is $\sum_{k=0}^{\infty} r(1-r)^{k}$, where $0<r<2$. Show more…

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Evaluate $\sum_{k=0}^{\infty} r(1-r)^{k}, 0<r<2.$

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Key Concepts

Sum Formula for an Infinite Geometric Series

When an infinite geometric series converges, its sum can be calculated using the formula S = a/(1 - r), where a is the first term and r is the common ratio. This formula provides a direct way to compute the total sum of all terms in the series.

Convergence Criteria for Infinite Series

For an infinite geometric series to converge, the absolute value of its common ratio must be less than one. This condition ensures that the successive terms decrease in magnitude sufficiently for the sum of the infinite series to approach a finite limit.

Geometric Series

A geometric series is a sequence of terms in which each term after the first is obtained by multiplying the previous term by a constant factor called the common ratio. This concept is central to understanding series that progress by constant multiplication, enabling the use of specific formulas to determine sums when the series converges.

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