What is the difference between these two sum notations?

Question

I often see sum-notation wirtten in one of the following ways:

AND:

Is there any difference between the two, or is it just a matter of style? thanks.

Answer 1

There is no difference in concept.

I like the first one better because of its format.

Answer 2

Well, like for many mathematical symbols, sometimes the two forms are used equivalently, other times the different layout is useful to help and distinguish different definitions of Sum.

While the first layout is almost generally reserved for the "ordinary" sum, the second is used by many people (including me) to indicate the sum in the meaning of Antidifference, computed for $i$ ranging from $1$ to $100$.
That is $$ f(x) = \Delta F(x) = F(x + 1) - F(x)\quad \Rightarrow \quad F(x) = \Delta ^{\, - 1} f(x) = \sum\nolimits_{\,x} {f(x)} $$ which for integral upper and lower limits returns almost the same value as the ordinary sum $$ \sum\nolimits_{\,i = 1\;}^{\,100} {f(i)} = F(100) - F(1) = \sum\limits_{i = 1}^{99} {f(i)} $$

But with the Antidifference interpretation, we can give a meaning also to a sum having the upper bound less than the lower one $$ \sum\nolimits_{\,i = 100\;}^{\;1} { = - f(i)} \sum\nolimits_{\,i = 1\;}^{\,100} {f(i)} $$ and, if $F(x)$ is defined also for real or complex values of the argument, then the sum has a meaning also for real/complex bounds $$ \sum\nolimits_{\,i = 1\;}^{\,y} {f(i)} = F(y) - F(1)\quad $$ so quite a different concept.