Prove the if the series converges then the collection additionally converges. Also, give an example to display that the converse is false, i.e., a instance in i m sorry converges however does not.

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

Proof. i think converges. Then we recognize

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. Thus, by the meaning of the limit, for every
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together that

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The first sum is a finite amount so that converges, and also the 2nd sum converges through comparison through . Hence, converges

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Counterexample. The converse is false. Let

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. Then,

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converges. However,

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diverges.


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Prove the if ∑ an converges climate ∑ (an)1/2 n-p converges because that p > 1/2
Prove the ∑ 1/an diverges if ∑ an converges

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A yes, really awesome book that I very recommend on just how to examine math and also be a math significant is Laura Alcock"s, How to examine as a mathematics Major:
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Basics: Calculus, direct Algebra, and Proof WritingCore mathematics Subjects.

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