Prove that $\sum_{n=1}^{+\infty}\frac{(2n+1)}{(n^2)(n+1)^2}=1$.

Question

Find $$\sum_{n=1}^{+\infty}\frac{(2n+1)}{(n^2)(n+1)^2}$$

The correct answer is 1 but I can't figure out how you end up to this conclusion

Answer 1

Since$$\frac{2n+1}{n^2(n+1)^2}=\frac1{n^2}-\frac1{(n+1)^2},$$your series is a telescoping series.