Prove that if the series converges then the series likewise converges. Also, offer an example to present that the converse is false, i.e., a instance in which converges however does not.

You are watching: If an converges then an^2 converges

*Proof.* Assume converges. Then we know

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The first sum is a finite amount so it converges, and the second sum converges by comparison via . Hence, converges

**Counterinstance.** The converse is false. Let

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converges. However,

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diverges.

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Prove that if

*∑ an*converges then

*∑ (an)1/2 n-p*converges for

*p > 1/2*

Prove that

*∑ 1/an*diverges if

*∑ an*converges

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