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