Trying to calculate the integral limit $\lim_{n\rightarrow\infty} \int_{-\sqrt n}^{\sqrt n}\left (1 - \frac{x^2}{2n}\right)^ndx$

Question

How to calculate following integral:

$$\lim_{n\rightarrow\infty}\int_{-\sqrt{n}}^{\sqrt{n}}{\left(1-\frac{x^2}{2n}\right)^n}dx$$ Prove that this integral exists and compute its value.

I just do not how start it. It is in real analysis, but I did not see any relation with real analsis.

Answer 1

The integrals may be written $\int_{-\infty}^\infty f_n(x)\, dx$, where $f_n(x) = 1_{[-\sqrt{n},\sqrt{n}]}(x)\cdot(1 - \frac{x^2}{2n})^n$. Now since $1 - \frac{x^2}{2n} \le e^{-x^2/2n}$, $|f_n(x)| \le e^{-x^2/2}$. Furthermore, $\lim_{n\to \infty} f_n(x) = e^{-x^2/2}$ and $\int_{-\infty}^\infty e^{-x^2/2}\, dx$ is finite. By the dominated convergence theorem, your limit is $\int_{-\infty}^\infty e^{-x^2/2}\, dx = \sqrt{2\pi}$.