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

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?

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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.

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