Hi transgalactic!

The standard method I know is to define .

Then your series corresponds to f(1/2).

If you integrate f(x) you should be able to find its limit (using the formula for a geometric series).

If you differentiate the result, you find f(x).

Substituting x=1/2 gives you your answer.

Alternatively, you can simply write out a number of terms.

Rearrange the terms in a triangle.

Find the sums of each row in the triangle.

And sum the results.