Find the sum of the series $\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\cdots$ My book directly writes-
$$\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\cdots=-\ln 2+1.$$
How do we prove this simply.. I am a high school student. 

In calculus there's this famous alternating harmonic series:
$S = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ... + (-1)^{(n+1)} \cdot \frac{1}{n} + ... $
(*) It's convergent and its sum is equal to $ln2$
Your series is equal to exactly $T = -S + 1$ so it's
also convergent and its sum must be exactly $(-ln2+1)$  
I realize that I didn't prove this (*) statement. I am not
aware of an elementary proof but there might be one.  
