Although the problem has been completely solved I'd like to comment that in general it can be proved that if

$a_k=\varphi(k+q)-\varphi(k),\; (q\in \mathbb{N}^*)\;\textrm{and}\; L=\lim_{k \to{+}\infty} \varphi(k)\in \mathbb{R}$

then,

$S=\displaystyle\sum_{k=1}^{+\infty}a_k=qL-\varphi(1)- \varphi (2)-\ldots - \varphi (q)$

In our case, the series can be inmediatly written in the form:

$S=-\displaystyle\sum_{k=1}^{+\infty}\left(\dfrac{1/2}{k+2}-\dfrac{1/2}{k}\right)$

that is, $q=2,\;\varphi(k)=1/2k$ . Then:

$S=-(2\cdot 0-\varphi(1)-\varphi(2))=\varphi(1)+\varphi(2)=\dfrac{1}{2}+\df rac{1}{4}=\dfrac{3}{4}$

Fernando Revilla