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.