(Inspired by a comment in answer <https://math.stackexchange.com/a/699264/442>.) 

**corrected** 

Let $\Psi(x) = \Gamma'(x)/\Gamma(x)$ be the [digamma function.](https://en.wikipedia.org/wiki/Digamma_function) Show
$$
\sum\_{n=1}^\infty (-1)^n\left(\Psi\left(\frac{n+1}{2}\right)
-\Psi\left(\frac{n}{2}\right)\right) = -1
$$
As noted, it agrees to many decimals. But care may be required since the convergence is only conditional.

**added**  

Both solutions are good. But could use explanation for exchange of summation.  

Note that
$$
2\sum\_{n=1}^\infty\sum\_{m=0}^\infty \left(\frac{(-1)^n}{2m+n}
-\frac{(-1)^n}{2m+n+1}\right)
=
2\sum\_{m=0}^\infty\sum\_{n=1}^\infty \left(\frac{(-1)^n}{2m+n}
-\frac{(-1)^n}{2m+n+1}\right)
= -1
$$
is correct. But "Fibini" justification fails, since
$$
\sum\_{m=0}^\infty\sum\_{n=1}^\infty \left|\frac{(-1)^n}{2m+n}
-\frac{(-1)^n}{2m+n+1}\right| = \infty
$$
Similarly in Random Variable's solution, the exchange $\sum\_{n=1}^\infty \int\_0^1 = \int\_0^1 \sum\_{n=1}^\infty$, although correct, cannot be justified by Fubini.