Consider the <em>k</em>-th partial sum,

More compactly,

(this is just another case of a similar sum you asked about a while ago [24494877])
The infinite sum is the limit of the partial sum as <em>k</em> goes to infinity. We have

since the non-constant terms in the limit converge to 0.
Alternatively, recall that for |<em>x</em>| < 1, we have

Differentiating both sides gives

also valid for |<em>x</em>| < 1. Take <em>x</em> = 1/<em>π</em> and you get the sum you want to compute.