Hi,(adsbygoogle = window.adsbygoogle || []).push({});

I can't find a way to prove the convergence of the following sum regarding to parametera:

[tex]

\sum_{n=1}^{ \infty } a_{n}

[/tex]

where

[tex]

a_{n} = \frac { \sqrt{1+ \frac{1}{n}} - 1}{n^{a}}

[/tex]

I already proved the neccessary condition for convergence, ie that

[tex]

\lim_{n \rightarrow \infty} a_{n} = 0

[/tex]

And it showed thatamust be in[itex](0, \infty) [/itex].

But I can't figure out how to prove the convergence. I triedd'Alembert'scriterion,comparing criterion,limite comparing criterionbut no gave me some useful result (withd'AlembertI got very complicated expression I wasn't able to simplify).

And one more question: in school I didn't understand, whether there is a equivalency inAbel-Dirichlet's criterionfor convergence. I mean if neither condition of the theorem is passed, could we say that the sum diverges?

Thank you.

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# Homework Help: How to prove the convergence?

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