Index of summation - integer? Can the index of summation of a sum be anything other than an integer? What is the reasoning behind the answer?

You will have obvious issues if you are attempting an uncountable sum of non-zero values, but otherwise there is no greater difficulty than sums over an integer index  
Here is an example:
$$\sum_{\frac{a}{b} \in \mathbb{Q} \cap (0,1]} \left(\frac{1}{b}\right)^k = \frac{\zeta(k-1)}{\zeta(k)}$$  where $\frac a b$ is a rational expressed in lowest terms and real $k \gt 2$
