Question

For what value of $r$ does $1+r+r^{2}+r^{3}+\cdots=10 ?$

   For what value of $r$ does $1+r+r^{2}+r^{3}+\cdots=10 ?$

Calculus: Early Transcendentals

Calculus: Early Transcendentals

William Briggs, Lyle… 3rd Edition

Chapter 10, Problem 90

Instant Answer

Solved by Expert Vipender Yadav

05/31/2020

Step 1

The sum of an infinite geometric series is given by the formula $S = \frac{a}{1-r}$, where $S$ is the sum, $a$ is the first term, and $r$ is the common ratio. This formula is valid only when $-1 < r < 1$. Show more…

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For what value of $r$ does $1+r+r^{2}+r^{3}+\cdots=10 ?$

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Key Concepts

Infinite Geometric Series

An infinite geometric series is a series of the form a + ar + ar² + ar³ + ... that continues indefinitely. It involves a starting term 'a' and a constant multiplier 'r' known as the common ratio, which is applied to generate each subsequent term.

Convergence of a Geometric Series

For an infinite geometric series to converge, the absolute value of the common ratio must be less than one (|r| < 1). This condition is essential because it ensures that the terms of the series diminish in size, allowing the sum of the series to approach a finite limit.

Sum Formula for an Infinite Geometric Series

When the 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 means of finding the total sum of all the terms in the series.

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