How do you write a sum in expanded form?

Perhaps you just mean to convert it from “summation form” (“sigma form”) to a written out form?

For something like ##sum_{i=1}^{n}i^{2}##, the summation symbol ##Sigma## just means to “add up”. Putting an ##i=1## underneath the summation symbol means to start the value of ##i## at 1. It is then assumed that ##i## keeps increasing by 1 until it reaches ##i=n##, where ##n## is the number above the summation symbol. The ##i^2## represents the formula for the terms that get added, first when ##i=1##, then ##i=2##, then ##i=3##, etc…, until ##i=n##.

Therefore, the answer would be ##sum_{i=1}^{n}i^{2}=1^{2}+2^{2}+3^{2}+cdots+(n-1)^2+n^2##.

This example is interesting in that there is a shortcut formula for adding up the first ##n## squares: it equals

##frac{n(n+1)(2n+1)}{6}=frac{1}{3}n^[3}+frac{1}{2}n^{2}+frac{1}{6}n.##

You should take the time to check that this works when, for instance, ##n=5##.