From the geometric series we can arrive at a couple of attractive-looking series:
For example, and .
Furthermore, from the sums and we obtain (after setting )
This last sum reminds me of the identity for positive integers , though appear in the sums in different places!
So for what values of are sums (1)-(4) valid (i.e. when do they converge)? For any positive integer the sum converges when , so this means sums (1), (3) and (4) converge when . In other words, , or equivalently, is closer to 0 than to -1. Hence the real part of must be at least -1/2. Similarly, sum (2) converges when , which is equivalent to the real part of being at least 1/2.