TITLE: What do I do in the case of this double series?
QUESTION [6 upvotes]: I am student who learns from examples, and I have yet to see what happens when two sums such as this one,
$$\displaystyle\sum\limits_{a=1}^2 \displaystyle\sum\limits_{b=a+1}^2 4\left[\left(\frac{1}{b-a}\right)^8 - \left(\frac{1}{b-a}\right)^4\right]$$
be solved. I am sure, when a = 1, the inner sum should solve itself like any "normal summation", but when a = 2, what happens? Do I skip the process of the inner summation; do I proceed with the inner sum, and if so, in what ways do I solve the inner sum; or do I solve this problem in a completely different way?

REPLY [2 votes]: Thinking inductively on the upper and lower indices, we can think of a summation as a difference of two functions on the integers:
$$
\sum_{k=a}^bf(k)=F(b)-F(a-1)
$$
Even in this generalization, if $b=a-1$, the sum is $0$ irregardless of $f$.