Evaluation of $\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)$? I would like to evaluate the sum
$$
\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)
$$
Where $\operatorname{Si}$ is the sine integral, defined as:
$$\operatorname{Si}(x) := \int_0^x \frac{\sin t}{t}\, dt$$
I found that the sum could be also written as
$$
-\sum\limits_{n=0}^\infty \int_n^\infty \frac{\sin t}{t}\, dt
$$
Anyone have any ideas?

One can find the following series,
$$ \displaystyle\sum_{n=0}^\infty \left(\mathrm{Si}(n)-\frac{\pi}{2}\right) = -1 - \frac{\pi}{4} - \sum _{k=0}^{\infty}{\frac { \left( -1 \right) ^{k}{\it 
B} \left( 2\,k+2 \right) }{ \left( 2\,k+1 \right) \Gamma 
 \left( 2\,k+3 \right) }} \approx -1.869201195, $$
where $B(n)$ are the Bernoulli numbers. 
