Find $\lim\limits_{n\to\infty}\sum\limits_{k=1}^n \frac{2k+1}{k^2(k+1)^2}$

Question

I have to find the limit $$\lim_{n\to\infty}\sum_{k=1}^n \frac{2k+1}{k^2(k+1)^2}.$$ I tried to make it into a telescopic series but it doesn't really work out...

$$\lim_{n\to\infty} \sum_{k=1}^n \frac{2k+1}{k^2(k+1)^2}=\sum_{k=1}^n \left(\frac{1-k}{k^2}+\frac1{k+1}-\frac1{(k+1)^2} \right)$$ so that is what I did using telescopic...

I said that:

$$\frac{2k+1}{k^2(k+1)^2}=\frac{Ak+B}{k^2}+\frac C{k+1}+\frac D{(k+1)^2}$$ but now as I look at it.. I guess I should "build up the power" with the ${k^2}$ too, right?

Answer 1

$$\lim_{n\rightarrow\infty}\sum^{n}_{k=1}\bigg[\frac{1}{k^2}-\frac{1}{(k+1)^2}\bigg]$$