Explain why or why not Determine whether the

Chapter 10, Problem 47E

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QUESTION:

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. A series that converges must converge absolutely.

b. A series that converges absolutely must converge.

c. A series that converges conditionally must converge.

d. If \(\sum a_{k}\) diverges, then \(\sum\left|a_{k}\right|\) diverges.

e. If \(\sum_{ }^{ }a_k^{\ 2}\) converges, then \(\sum a_{k}\) converges.

f. If \(a_{k}>0\) and \(\sum a_{k}\) converges, then \(\sum_{ }^{ }a_k^{\ 2}\) converges.

g. If \(\sum a_{k}\) converges conditionally, then \(\sum\left|a_{k}\right|\) diverges.

Questions & Answers

QUESTION:

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. A series that converges must converge absolutely.

b. A series that converges absolutely must converge.

c. A series that converges conditionally must converge.

d. If \(\sum a_{k}\) diverges, then \(\sum\left|a_{k}\right|\) diverges.

e. If \(\sum_{ }^{ }a_k^{\ 2}\) converges, then \(\sum a_{k}\) converges.

f. If \(a_{k}>0\) and \(\sum a_{k}\) converges, then \(\sum_{ }^{ }a_k^{\ 2}\) converges.

g. If \(\sum a_{k}\) converges conditionally, then \(\sum\left|a_{k}\right|\) diverges.

ANSWER:

SOLUTIONStep 1(a). A series that converges must converge absolutely.This statement is false.A series is said to be convergent if the sequence of its partial sums tends to a limit,ie the partial sums become closer and closer to a given number when the number of their terms increases.A series is said to be absolutely convergent if and is convergent.If is convergent and is divergent then the series is called conditionally convergent.

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