$$S=\sum_{k=1}^\infty\left(\frac{k}{k+1}\right)^{k^2};\hspace{10pt}S_n=\sum_{k=1}^n\left(\frac{k}{k+1}\right)^{k^2}$$ Let $S_n$ represent its partial sums and let $S$ represent its value. Prove that $S$ is finite and find an n so large that $S_n$ approximate $S$ to three decimal places.
Solution: first of all, I think that we Will use L'hopital rule and then use root test while starting to solve this. But how?
Hint: use the $k$th root test.
If $a_k$ are the coefficients, that is:
$$a_k=\left(\frac{k}{k+1}\right)^{k^2}$$
then:
$$\sqrt[k]{a_k}=\left(\frac{k}{k+1}\right)^{k}=\left(1-\frac{1}{k+1}\right)^k$$
With some fantasy, do you recognize this sequence? Does it converge to something you know?