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Chapter 7: Q. 69 (page 654)

Prove that if the series $\sum_{k = 1}^{\infty} a_{k}$ diverges, then the series $\sum_{k = 1}^{\infty} |a_{k}|$ also diverges.

Short Answer

Expert verified

The series $\sum_{k = 1}^{\infty} a_{k}$ is convergent

The series $\sum_{k = 1}^{\infty} |a_{k}|$ is divergent

Step by step solution

01

Step 1. Given information

$\sum_{k = 1}^{\infty} |a_{k}|$ and $\sum_{k = 1}^{\infty} a_{k}$ are the given series

02

Step 2. Finding whether ∑k=1∞ akis convergent

Assume that $\sum_{k = 1}^{\infty} |a_{k}|$ is not divergent,

Therefore, $\sum_{k = 1}^{\infty} |a_{k}|$ is convergent.

If $\sum_{k = 1}^{\infty} a_{k}$ is convergent but it is given that it is divergent.

If the series $\sum_{k = 1} a_{k}$ is absolutely convergent, then the series $\sum_{k = 1} a_{k}$ is convergent.

Therefore, $\sum_{k = 1}^{\infty} a_{k}$ is convergent, which is a contradiction as it is given that the series $\sum_{k = 1}^{\infty} a_{k}$ is divergent.

Therefore, the supposition that the series $\sum_{k = 1}^{\infty} |a_{k}|$ is not divergent is wrong.

Hence, the series $\sum_{k = 1}^{\infty} |a_{k}|$ is divergent.

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Most popular questions from this chapter

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.

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

Prove Theorem 7.31. That is, show that if a function a is continuous, positive, and decreasing, and if the improper integral $\int_{1}^{\infty} a (x) d x$ converges, then the nth remainder, $R_{n}$ , for the series $\sum_{k = 1}^{\infty} a (k)$ is bounded by $0 \leq R_{n} = \sum_{k = n + 1}^{\infty} a (k) \leq \int_{n}^{\infty} a (x) d x$

Let f(x) be a function that is continuous, positive, and decreasing on the interval $[1, \infty)$ such that role="math" localid="1649081384626" $\underset{x \to \infty}{l i m} f (x) = α > 0$ . What can the divergence test tell us about the series $\sum_{k = 1}^{\infty} f (k)$ ?

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.

$0.305130513051 . . .$

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 $0$ .

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