Which is the easiest way to find the value of $\sum \limits_{n=1}^k \frac{(-1)^{n-1}\times 2n}{{(2n-1)\times (2n+1)}} $
There is a hint, which says we can use $ {{1 \over 2}\over{r-1}} + {{1 \over 2}\over{r+1}} = {r \over{(r-1) (r+1)} } $
but I am not sure how, any ideas guys?