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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

If $\lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty a n d \sum_{k = 1}^{\infty} a_{k}$ converges, explain why we cannot draw any conclusions about the behavior of $\sum_{k = 1}^{\infty} b_{k}$ .

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

$\sum_{k = 1}^{\infty} {(\frac{k}{1 + k})}^{k^{2}}$

Use any convergence test from this section or the previous section to determine whether the series in Exercises 31–48 converge or diverge. Explain how the series meets the hypotheses of the test you select.

$\sum_{k = 1}^{\infty} \frac{\sqrt{k}}{k^{2} + 3}$

Prove Theorem 7.24 (a). That is, show that if c is a real number and $\sum_{k = 1}^{\infty} a_{k}$ is a convergent series, then $\sum_{k = 1}^{\infty} c a_{k} = c \sum_{k = 1}^{\infty} a_{k}$ .

Determine whether the series $\sum_{k = 0}^{\infty} {(\frac{- 9}{8})}^{k}$ 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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