I am looking at two series: I am trying to determine whether they converge or diverge

$$\sum_{n=1}^\infty \cos\left(\frac{1}{n^2}\right) $$

and $$\sum_{n=1}^\infty \left(\frac{n}{n+1}\right)^{n^2} $$

For the first one, I am pretty sure it diverges, but I am having trouble finding what test to use.

The second one, I have no clue really. Can you help?

  • $\begingroup$ $n$-th term test for the first one. $\endgroup$ – vadim123 Dec 3 '15 at 16:23
  • $\begingroup$ Oh, obviously, I was trying to use another test. Thanks. Any idea for the second one? $\endgroup$ – Goose719 Dec 3 '15 at 16:27

For the first , $\displaystyle \lim_n\cos\left(\frac{1}{n^2}\right)=1\not=0$ , so divergent.

For the second ,(use Cauchy root test). Let , $a_n=\left(\frac{n}{n+1}\right)^{n^2}$. Then $a_n^{1/n}\to1/e<1$. So convergent.


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