Convergence of $ \begin{array}{l}\sum _{n=0}^{\infty }\frac{\left(2k^2+2k+1\right)}{k^4+2k^3+k^2}\end{array}$

Question

Prove that :$ \begin{array}{l}\sum _{n=0}^{\infty }\frac{\left(2k^2+2k+1\right)}{k^4+2k^3+k^2}\end{array}$ where $k=2n+1$ converges. Also find the value it converges to.

Answer 1

Hint:

Use partial fraction decomposition.

$$\sum_{n=0}^{\infty} \frac{1}{k^2} +\frac{1}{(k+1)^2} = \sum_{n=1}^{\infty} \frac{1}{n^2} = ?$$

Answer 2

HINT:

$$\frac{2 (2 n+1)^2+2 (2 n+1)+1}{(2 n+1)^4+2 (2 n+1)^3+(2 n+1)^2}=\frac{1}{(2n+1)^2}+\frac{1}{4(n+1)^2}\tag1$$