Evaluating the infinite sum $\lim_{n \to \infty}\sum_{r=0}^n\frac{1}{2r^2 +3r+1}$

Question

How do I evaluate :

$\lim_{n \to \infty}\displaystyle \sum_{r=0}^n\left(\dfrac{1}{2r^2 +3r+1}\right)$

I tried sandwhich theorem and even wrote down some terms to see any particular pattern but these methods didn't help.

The thing to be summed can also be written as:

$2(\frac 1 {1+2r}- \frac1{2(r+1)})$

Answer 1

$$\sum_{r\ge 0}\frac{1}{(r+1)(2r+1)}=2\sum_{r\ge 0}(\frac{1}{2r+1}-\frac{1}{2r+2})=2(1-\frac{1}{2}+\cdots)=2\ln 2=\ln 4$$