TITLE: Examples of problems that are easier in the infinite case than in the finite case.
QUESTION [119 upvotes]: I am looking for examples of problems that are easier in the infinite case than in the finite case. I really can't think of any good ones for now, but I'll be sure to add some when I do.

REPLY [2 votes]: I don't believe nobody mentioned applications of the Central Limit Theorem (CLT).
For example calculating this:
\begin{align}
\int^{n+\sqrt[]{n}}_0 \frac{x^{n-1}}{(n-1)!}e^{-t}\,dt 
\end{align}
is very easy when $n\to \infty$ using the CLT.
And yet another application of CLT itself is Wilk's Theorem very strong statement  about the infinity case.