# Finding the nth partial sum of a telescoping series

$$\sum _{n=1}^{\infty }\:\left[\frac{2}{\left(n+1\right)}-\frac{2}{\left(n+3\right)}\right]$$

This question is off from webwork and I already got everything right except for finding the nth partial sum. Here is what I have:

s3 = 1 + 2/3 - 2/5 - 2/6

s4 = 1 + 2/3 - 2/6 - 2/7

s5 = 1 + 2/3 - 2/7 - 2/8

It converges to 1 + 2/3

Now my nth partial sum is: 1 + 2/3 - 2/(n+1) - 2/(n+2)

But it seems to be incorrect. Can someone explain what I'm doing wrong?

When you sum up to $$n$$ the surviving negative terms are $$\frac 2{n+2}$$ and $$\frac 2{n+3}$$. See this and this for $$n=10$$. You are off by $$1$$ in the denominators.