Good typography relies on the balance of all aspects of black and white on the page. Uniformly spaced lines for the ordinary text make for good legibility. That's why some symbols, that in display are rendered with limits above and below, are set with limits on the side when used in in-line formulas.
I too, when I began to use TeX, tried to set limits for sums above and below also in in-line formulas, but I soon realized that it's wrong: two white bands separated that line from the next ones.
For that very reason the symbols for summation and integral are set in two different sizes: a smaller one for in-line formulas, a larger one for displayed formulas that don't have spacing constraints.
Displayed formulas are set using (automatically)
would produce the same symbol used in displayed formulas and set limits above and below. But this will damage the balance of the page beyond repair. A less invasive construction
and its sibling
will do less damage to the page, but will nevertheless spoil it.
Such constructions have their use, for examples in tables where TeX would use in-line math mode. But I will never suggest to use them in normal text.
$$...$$, see: tex.stackexchange.com/questions/503/why-is-preferable-to
\displaystyle(but not the larger size) you might want to rephrase the question...in that case you might want to look at