Examples of problems that are easier in the infinite case than in the finite case. 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.
One can compute the value of 
$$\int_{0}^{\infty}e^{-t^{2}}\,\mathrm{d}t$$
exactly. This is known as the Gaussian integral, and it has its own Wikipedia page. The answer turns out to be $\frac{1}{2}\sqrt{\pi}.$

But one cannot do the same with
$$\int_{0}^{x}e^{-t^{2}}\,\mathrm{d}t$$
because the antiderivative of the integrand is not an elementary function. This is why we gave a name to the error function $$\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}\,\mathrm{d}t,$$ which also has its own Wikipedia page.

In that sense, the infinite case is easier than the finite case.



Addendum: The same phenomenon occurs for variants of this integral, in particular we can transform the integrand to evaluate $$\int_{-\infty}^{\infty}ae^{-(t-b)^{2}/(2c^{2})}\,\mathrm{d}t = \sqrt{2}a\lvert c\rvert \sqrt{\pi}$$ as detailed here on Wikipedia.
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.