Limit $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n^2}{(n^2+k^2)(n+k)}$

Question

Calculate the limit
$$\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n^2}{(n^2+k^2)(n+k)}$$

---

I tried solving it by changing it a Riemann sum then integrating, however I couldn't manipulate the algebra to its form. Is there another way of doing this or am I on the right track?

Answer 1

HINT:

$\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n^2}{(n^2+k^2)(n+k)}=\lim_{n\to\infty}\frac1n\sum_{k=1}^{n}\frac1{(1+(k/n)^2)(1+k/n)}$

Now, $\displaystyle\lim_{n\to\infty}\frac1n\sum_{k=1}^{n}f(k/n)=\int_0^1f(x)dx$

Please let me know if you can complete the task after using Partial Fraction Decomposition