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.

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Proof. Assume converges. Then we know . Therefore, by the definition of the limit, for all 0" title="Rendered by QuickLaTeX.com" height="11" width="36" style="vertical-align: 0px;"/> tright here exists an integer such that " title="Rendered by QuickLaTeX.com"/>

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 . Then, " title="Rendered by QuickLaTeX.com"/>

converges. However, " title="Rendered by QuickLaTeX.com"/>

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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Basics: Calculus, Linear Algebra, and Proof WritingCore Mathematics Subjects.

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