Examples of use of integrals to show convergence of a limit

Recently I got the problem where I had to show the existence and finiteness of the limit of $$a_n$$ where $$a_n$$ is bounded and such that both limits \begin{align} \lim_{n\to\infty} \cos(a_nt) \ \ \ \text{ and } \ \ \ \lim_{n\to\infty} \sin(a_nt) \end{align} exist for all $$t\in\mathbb R$$. A reasonable proof would be to argue that there can only by one accumulation point, because more would lead to a contradiction. It is not so hard, but one needs to think a little bit.

What did I do? Well, MSE as you guys know is full integrals. They are everywhere! Some see it as recreational math, but some see it as ways to proof things in an elegant manner. It is already done in here by Terry Tao. Also in this post proving $$e<\pi$$ and finally in this post where there was integral milking which actually lead to a proof of a serious result.

So to tackle my problem with sines and cosines, I have used the integrals \begin{align} \int_\mathbb R \frac{\cos(a_nx)}{x^2+1}\,dx \ \ \ \text{ and } \ \ \ \int_\mathbb R\frac{\sin(a_nx)}{x(x^2+1)}\,dx \end{align} together with Dominated Convergence Theorem to prove the convergence of $$a_n$$. The original problem and the detailed solution can be found here.

How did I came up with these integrals? I wanted to use the fact that is pretty dumb if $$a_n$$ would not converge. Indeed, it would lead to ridiculous results. Moreover since the cos and the sin are bounded, I thought there must be a nice integral....before I finished my sentence in my head those integrals popped up with a clean button which said "DCT"!

The first one is something everyone has seen in Complex Analysis in the chapter "contour integration" and the second one as well. However the first one actually comes up naturally in probability (cauchy distribution) while the second one is to exercise (contour) integration, say. I believe they are constantly posted in MSE, so I couldn't miss it.

Question. What are examples where the use of integrals (or series) provide a proof of existence/non-existence of a limit?

This question makes sense, because it shows that the fancy recreational integrals can actually be beneficial for proofs. Integrals connect functions with each other. In this specific case they connected the periodic cosine with the exponential function which is invertible. I think there might be a lot more of such examples.

Of course series are also allowed, since one can write series as an integral anyway. By the way I do not mean to prove really really trivial results, but I cannot put a line there...

• @James that is what I want to say yes – Shashi Feb 15 at 10:21

The integral comparison theorem states that $$\lim\limits_{n\to \infty}\sum\limits_{m=0}^n a_n<\infty \Longleftrightarrow \lim\limits_{n\to \infty}\int_0^n a_x dx<\infty.$$
$$\lim \limits_{n\to \infty}\int_1^n \frac{1}{x}dx=\lim \limits_{n\to \infty}\log(n)-\log(1)=\lim \limits_{n\to \infty}\log(n)=\infty,$$ so $$\sum\limits_{n=1}^\infty \frac{1}{n}=\infty.$$
• It is something like this we want to know the convergence of $a_n$ and we have somewhere an integral $I_a=\int f_a(t) \, dt$ with certain properties. The sequence $I_{a_n}$ should then say something about the sequence $a_n$. Did you see the example I provided? math.stackexchange.com/a/3112599/349501 – Shashi Feb 15 at 13:32