TITLE: Does the series $1-\frac12+\frac12-\frac1{2^2}+\frac13-\frac1{2^3}+\frac14-\frac1{2^4}+\frac15-\frac1{2^5}+\cdots$ converge or diverge?
QUESTION [9 upvotes]: $1-\frac{1}{2}+\frac{1}{2}-\frac{1}{2^2}+\frac{1}{3}-\frac{1}{2^3}+\frac{1}{4}-\frac{1}{2^4}+\frac{1}{5}-\frac{1}{2^5}+\cdots$

I've be trying to figure out how to write this series symbolically so I can examine it's limit, but I'm having trouble.  So far the best I've come up with is:
$\sum_{n=0}^{\infty}(-1)^n\left(\frac{1}{2^n}-\frac{1}{2^{n+1}}\right)$
But obviously the above does not properly reproduce the series.

REPLY [5 votes]: Observe that $$S_{n}=\displaystyle\sum_{i=1}^\left\lceil \frac{n}{2}\right\rceil\left(\dfrac{1}{i}\right)-\displaystyle\sum_{i=1}^\left\lfloor \frac{n}{2}\right\rfloor\left(\dfrac{1}{2^i}\right)\ge\left(\displaystyle\sum_{i=1}^\left\lceil\frac{n}{2}\right\rceil\dfrac{1}{i}\right)-1$$ Since the sequence of partial sum diverges, the series also diverges.