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?

The "index" of summation, the way I'm interpreting your question, corresponds to an enumeration of the elements of the set over which you are summing. For example, if you want to add the numbers $x_{1},...,x_{n}$, then you would write
$$\sum_{k=1}^{n}x_{k}$$
You can also think of this as
$$\sum_{x\in S}x$$
where $S=\{x_{1},...,x_{n}\}$.
However, there are sets which are not countable - this means the elements of the set are not in one-to-one correspondence with the set of integers. For example, you can take the sum of every real number in $[0,1]$, which you could write as
$$\sum_{x\in [0,1]}x$$
The problem is that because this set is uncountable, so there's no natural ordering for which to add these numbers. And in this case the sum is infinite. However if you know that a sum
$$\sum_{x\in S}x<\infty$$
then, even if $S$ is uncountable, this can be rewritten as a sum over a countable set.
