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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. Therefore, by the definition of the limit, for all
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such that

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

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Counterinstance. 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 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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A really awesome book that I extremely recommend on exactly how to examine math and be a math major is Laura Alcock"s, How to Study as a Mathematics Major:
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Basics: Calculus, Linear Algebra, and Proof WritingCore Mathematics Subjects.

See more: Letra De Los Peces En El Rio, Los Peces En El Rio: Canción Con Letra


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