Question

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 8\left(\frac{1}{3}\right)^{k-1} $$

Name: Problem 62, Chapter 11: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2020-06-17T19:39:55-08:00
Duration: 1 min 33 s
Channel: Dale Sanford
Description: Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. ∑k=1^∞ 8((1)/(3))^k-1

   Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
\sum_{k=1}^{\infty} 8\left(\frac{1}{3}\right)^{k-1}
$$

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan 4th Edition

Chapter 11, Problem 62

Instant Answer

Solved by Expert Dale Sanford

06/17/2020

Step 1

The common ratio, r, is given by $\frac{1}{3}$ in this series. Since $\frac{1}{3}$ is between -1 and 1, the series converges. Show more…

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Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 8\left(\frac{1}{3}\right)^{k-1} $$

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Key Concepts

Geometric Series

A geometric series is a series where each term is obtained by multiplying the previous term by a constant called the common ratio. The general form can be expressed as a + ar + ar^2 + ar^3 + ... , and it is foundational in understanding sequences and series in mathematics.

Convergence Criteria for Infinite Geometric Series

An infinite geometric series converges when the absolute value of the common ratio is less than 1 (|r| < 1). This criterion ensures that the terms of the series become sufficiently small as the sequence progresses, allowing the series to approach a finite limit rather than diverging to infinity.

Sum Formula for Convergent Infinite Geometric Series

If 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 method to find the limit of the partial sums as the number of terms increases indefinitely.

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