Find the sum of the series $\sum_{n=1}^{\infty} \frac{2n+3}{n(n+1)(n+2)}$ 
Find the sum of the series $\sum_{n=1}^{\infty} \frac{2n+3}{n(n+1)(n+2)}$

My attempt: 
I tried partial fractions decomposition and I get :
$$\sum_{n=1}^{\infty} \frac{2n+3}{n(n+1)(n+2)}=\sum_{n=1}^{\infty}\frac {3}{2n}-\frac 1{n+1}-\frac 1{2(n+2)}=\frac{1}{2}\sum_{n=1}^{\infty}\frac 3{n}-\frac 2{n+1}-\frac 1{n+2}.$$
This should be the solution but the partial sum sequence I can't figure out the formula... what should I do in this case?

Hint: The last sum can be written as $$\sum_{n=1}^\infty\left[2\left(\frac1n-\frac1{n+1}\right)+\left(\frac1n-\frac1{n+2}\right)\right]$$
The is a telescoping sum and we can expicitly write down the sequence of partial sums.
