TITLE: What is $\sum_{r=1}^\infty\frac{r+2}{2^{r+1}(r)(r+1)}$?
QUESTION [5 upvotes]: Find out the sum of the following infinite series
$$\frac{3}{2^2(1)(2)} + \frac{4}{2^3(2)(3)} +\dots+\frac{r+2}{2^{r+1}(r)(r+1)}+\cdots
$$
up to $r\to\infty$.
MY TRY:-  I tried to split $r+2$ as $[(r+1) +{(r+1)-r}]$ so that I can cancel one term from each terms in the numerator. Then I got an expression which was like Harmonic-Geometric series. But I could not do further any more after this.

REPLY [9 votes]: If you do the partial fraction expansion, the summand becomes
$$\frac{1}{2^{r+1}}\left( \frac{2}{r} - \frac{1}{r+1}\right) = \frac{1}{2^r r} - \frac{1}{2^{r+1}(r+1)},   $$
So the sequence is telescoping.  All terms cancel except the first and so sum equals $\frac12$.