Do I use partial fractions? How to set it up?

Question

$$\sum_{n=1}^\infty \left(\frac{1}{n} - \frac{1}{n+2}\right)$$

Answer 1

This is a telescoping series whose sum is 3/2.

Answer 2

Your left side has no free variables-$n$ is a dummy that you sum over. The right side has $n$ free, so they cannot be equal. The left side is either a number (here $\frac 32)$ or nonsense. The point of the comments about telescoping series is that most of the terms cancel. What is left?

Answer 3

You don't want to use partial fractions here. Look at the partial sums: $$ s_k=\sum_{n=1}^k \left(\frac{1}{n}-\frac{1}{n+2} \right) $$

Try to come up a closed formula for $s_k$ (something that depends upon just $k$) and then look at

$$ \lim_{k\rightarrow\infty}s_k. $$

Good luck!