How do you determine whether the infinite sequence $a_n=ncos(npi)$ converges or diverges?

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Answer 1 · 2014-09-24T11:08:15

The sequence $a_n=ncos(n pi)$ diverges.

Let us look at some details.

Let us take the limit of its subsequence of only even terms.

$\lim_{n to infty}a_{2n}=lim_{n to infty}(2n)cos(2npi)$

since $cos(2npi)=1$ for all integer $n$ ,

$=lim_{n to infty}(2n)=2(infty)=infty$

Since the subsequence diverges, the original sequence also diverges.

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