Sum $\sum_{n=1}^{\infty} \frac{1}{n}-\frac{1}{n+2}$ I have to first prove whether the following series is convergent, and then find its limit if it exists:
$\sum_{n=1}^{\infty} \frac{1}{n}-\frac{1}{n+2}$
So, I have proved that it is convergent, but I'm having trouble in finding the limit. Can someone help me out?

The ans is $\frac{3}{2}$. Compute the sequence of partial sum and you'll figure it out.
Spoiler :
$S_n = (1+\frac{1}{2}+...+\frac{1}{n})-(\frac{1}{3}+\frac{1}{4}+...+\frac{1}{n+2})$
$= (1+\frac{1}{2}+...+\frac{1}{n})+(1+\frac{1}{2})-(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{n+2})$
$=(\gamma_{n} + ln {n}) + \frac{3}{2} - (\gamma_{n+2} + ln (n+2))$
$=(\gamma_n - \gamma_{n+2}) +ln {\frac{n}{n+2}}+ \frac{3}{2}$
Therefore $\lim_{n\to \infty} S_n = \frac{3}{2}$
