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Chapter 7: Q. 72 (page 654)

Changing the order of the summands in a conditionally convergent series can change the value of the sum. We showed this earlier in the section for the alternating harmonic series

Short Answer

Expert verified

Series is divergent.

Step by step solution

Step 1. Given information

Changing the order of the summands in a conditionally convergent series can change the value of the sum.

Explanation

Step 3. proof

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Most popular questions from this chapter

Let $a : [1, \infty) \to ℝ$ be a continuous, positive, and decreasing function. Complete the proof of the integral test (Theorem 7.28) by showing that if the improper integral $\int_{1}^{\infty} a (x) d x$ converges, then the series localid="1649180069308" $\sum_{k = 1}^{\infty} a (k)$ does too.

Let $\sum_{k = 0}^{\infty} c r^{k}$ and $\sum_{k = 0}^{\infty} b v^{k}$ be two convergent geometric series. If b and v are both nonzero, prove that $\sum_{k = 0}^{\infty} \frac{c r^{k}}{b v^{k}}$ is a geometric series. What condition(s) must be met for this series to converge?

True/False:

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If $a_{k} \to 0$ , then $\sum_{k = 1}^{\infty} a_{k}$ converges.

(b) True or False: If $\sum_{k = 1}^{\infty} a_{k}$ converges, then $a_{k} \to 0$ .

(c) True or False: The improper integral $\int_{1}^{\infty} f (x) d x$ converges if and only if the series $\sum_{k = 1}^{\infty} f (k)$ converges.

(d) True or False: The harmonic series converges.

(e) True or False: If $p > 1$ , the series $\sum_{k = 1}^{\infty} k^{- p}$ converges.

(f) True or False: If $f (x) \to 0$ as $x \to \infty$ , then $\sum_{k = 1}^{\infty} f (k)$ converges.

(g) True or False: If $\sum_{k = 1}^{\infty} f (k)$ converges, then $f (x) \to 0$ as $x \to \infty$ .

(h) True or False: If $\sum_{k = 1}^{\infty} a_{k} = L$ and ${S_{n}}$ is the sequence of partial sums for the series, then the sequence of remainders ${L - S_{n}}$ converges to $0$ .

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

$0.99999 . . .$

Determine whether the series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ converges or diverges. Give the sum of the convergent series.

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Changing the order of the summands in a conditionally convergent series can change the value of the sum. We showed this earlier in the section for the alternating harmonic series

Short Answer

Step by step solution

Step 1. Given information

Explanation

Step 3. proof

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Most popular questions from this chapter

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