How to find sum of the infinite series $\sum_{n=1}^{\infty} \frac{1}{ n(2n+1)}$ $$\frac{1}{1 \times3} + \frac{1}{2\times5}+\frac{1}{3\times7} + \frac{1}{4\times9}+\cdots $$
How to find sum of this series?
I tried this: its $n$th term will be = $\frac{1}{n}-\frac{2}{2n+1}$; after that  I am not able to solve this.

Your series can be written as 
$$
    \frac{H_{x}}{2x} = \sum_{k=1}^\infty \frac{1}{k\cdot2(k+x)}\, ,
$$
with $x=\frac12$. $H_{\frac12}$ is given here: Harmonic numbers.
$$
    H_{\frac{1}{2}} = 2 -2\ln{2},
$$
which is your result...
