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Chapter 7: Q 60. (page 615)

Determine whether the series $\frac{99}{40} - \frac{99}{20} + \frac{99}{10} - \frac{99}{5} + 鈥�$ converges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series $\frac{99}{40} - \frac{99}{20} + \frac{99}{10} - \frac{99}{5} + 鈥�$ diverges.

Step by step solution

01

Step 1. Given information.

Given a series $\frac{99}{40} - \frac{99}{20} + \frac{99}{10} - \frac{99}{5} + 鈥�$ .

02

Step 2. Find if the series converges or not.

The standard form of a geometric series is ${鈭�}_{k = 0}^{鈭�} c r^{k}$ .

The geometric series converges if and only if $|r| < 1$ .

In the series role="math" localid="1648983721211" $\frac{99}{40} - \frac{99}{20} + \frac{99}{10} - \frac{99}{5} + 鈥�$ it can be seen that $c = \frac{99}{40}$ .

Every term after that is $- 2$ times the previous term.

It follows that $r = - 2$ .

Since $r = - 2$ , the series $\frac{99}{40} - \frac{99}{20} + \frac{99}{10} - \frac{99}{5} + 鈥�$ diverges.

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Most popular questions from this chapter

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

${鈭�}_{k = 1}^{鈭�} 鈥� (3 k 鈭� 1)^{鈭� 2 / 3}$

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$0.199999 . . .$

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$3.454545 . . .$

Prove that if ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges to L and ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges to M , then the series ${鈭�}_{k = 1}^{鈭�} (a_{k} + b_{k}) = L + M$ .

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$0.6345345 . . .$

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