Need hint: Determine whether the series $\sum_{n=1}^{\infty}(\frac{n}{n+1})^{n^2}$ is convergent or divergent.

Solution 1:

Let $u_n$ = $ \Big ( \frac {n}{n+1} \Big)^{n^2}$ , then since $ \ \lim_{n \to \infty}\sqrt[n]{({\frac{n}{n+1}})^{n^2}} = \lim_{n \to \infty}(\frac{n}{n+1})^n = \lim_{n \to \infty}(\frac{1}{1+\frac{1}{n}})^n$ = $\frac1e$ $ < 1 $ , so by Cauchy's root test the series is convergent .

Solution 2:

Note the following: $$\Big(\frac{n}{n+1}\Big)^{n}=\Big(\frac{m-1}{m}\Big)^{m-1}=(1-1/m)^{m-1}$$ That formula/function seems familiar, doesn't it?

Thereafter, note that: $$\Big(\frac{n}{n+1}\Big)^{n^{2}}={\Big(\Big(\frac{n}{n+1}\Big)^{n}\space\Big)}^{n}$$ Hint: Simple comparison test with a sequence you know well.

Solution 3:

Hint: Try with Cauchy's root test.