/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 Evaluate \(\sum_{k=1}^{\infty} \... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 16

Evaluate \(\sum_{k=1}^{\infty} \frac{8}{5^{k}}\).

Short Answer

Expert verified
The sum of the infinite geometric series \(\sum_{k=1}^{\infty} \frac{8}{5^{k}}\) is 2. This is because the first term is \(\frac{8}{5}\), the common ratio is \(\frac{1}{5}\), and the series converges. Using the formula for the sum of an infinite geometric series, S = \(\frac{a_1}{1 - r}\), and simplifying the expression, the sum is determined to be 2.

Step by step solution

01

Identify the first term and the common ratio

The given series is \(\sum_{k=1}^{\infty} \frac{8}{5^{k}}\). To find the first term, substitute k = 1: a_1 = \(\frac{8}{5^1}\) = \(\frac{8}{5}\). The common ratio can be identified from the terms of the series: r = \(\frac{1}{5}\)
02

Check if the series converges

For an infinite geometric series to converge, the common ratio must be between -1 and 1: -1 < r < 1 In our case, the common ratio is \(\frac{1}{5}\), which lies between -1 and 1, so the series converges.
03

Use the infinite geometric series formula

Now that we have confirmed that the series converges, we can use the formula for the sum of an infinite geometric series: S = \(\frac{a_1}{1 - r}\) Substitute a_1 = \(\frac{8}{5}\) and r = \(\frac{1}{5}\): S = \(\frac{\frac{8}{5}}{1-\frac{1}{5}}\)
04

Simplify the expression

First, simplify the denominator of the expression: 1 - \(\frac{1}{5}\) = \(\frac{5 - 1}{5}\) = \(\frac{4}{5}\) Next, divide the numerator by the simplified denominator: S = \(\frac{\frac{8}{5}}{\frac{4}{5}}\) To divide fractions, multiply the numerator by the reciprocal of the denominator: S = \(\frac{8}{5} * \frac{5}{4}\)
05

Evaluate the series

Now, multiply the fractions to find the sum of the series: S = \(\frac{8 * 5}{5 * 4}\) = \(\frac{40}{20}\) Simplify the fraction: S = 2 The sum of the given infinite geometric series is 2.

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