What are some good integration problems where you can use some of the function convergence theorem of Lesbegue integrals?

Question

I have learned about two major convergence theorem for the Lesbegue Measure:

• The monotone convergence theorem
• The dominated convergence theorem

These are useful theorems for calculating integrals such as

\begin{equation} \lim_{n \rightarrow \infty} \int_0^n (1 - \frac{x}{n})^n \text{log}(2 + \text{cos}(\frac{x}{n}))\text{dx} \end{equation}

What are some other good calculation problems?

Answer 1

You can derive interesting formulas for the Gamma function by writing $$\Gamma(x+1) = \int_0^\infty e^{-t} t^x \, dx = \lim_{n \to \infty} \int_0^n \left( 1 - \frac tn \right)^n t^x \, dt = \lim_{n \to \infty} \frac{n^x n!}{(x+1)\cdots(x+n)}.$$