Prove that $s$ is finite and find and an $n$ so large that $S_n$ approximates $s$ to 3 decimal places. 
Let $S_n$ represent its partial sum, and let $s$ represent its value.
  Prove that $s$ is finite and find and an $n$ so large that $S_n$
  approximates s to 3 decimal places.
  $$\sum_{k=1}^{\infty}\left(\frac{k}{k+1}\right)^{k^2}$$

Solution we use root test i think. But how to calculate this? Thanks. 

Hint: Let your series be $\sum_{1}^\infty a_k$.
Let $b_k=\left(\frac{k+1}{k}\right)^k$. Note that the sequence $(b_k)$ is increasing and has limit $e$. But we don't need to know that, it is enough to observe that $b_k \ge 2$.
We have $a_k =\left(\frac{1}{b_k}\right)^k$. Thus  $a_k\le \frac{1}{2^k}$. So we can bound the truncation error by the tail of a familiar geometric series.
For just showing that the series converges, Comparison with $\sum \fac{1}{2^k}$ does the job. If you want to use one of Ratio Test or Root Test, both will work, but Root Test is simpler. 
