Definite integral: $\lim\limits_{n\to\infty}\int_{-\sqrt n}^{\sqrt n}\left(1-\frac{x^2}{2n}\right)^n \,\mathrm dx$

Question

$$\lim_{n\to\infty}\int_{-\sqrt n}^{\sqrt n}\left(1-\frac{x^2}{2n}\right)^n \,\mathrm dx $$

I tried very hard to solve this integral and it looks like while $n$ goes to infinity it will give us a Gaussian distribution (I feel somehow, I mean maybe it is obvious from inside of integral).

Anyway, I tried to split this function inside of integral but it didn't work.

Computer the integral first by substituing $\frac{x}{\sqrt(2n)}$ =sin x. — GTX OC, Oct 26 '13 at 16:24
This and this are declared duplicates of each other, but not duplicates of your post. The answers in both however are really good and pertinent here, and so I thought you might enjoy them. — J. W. Perry, Oct 26 '13 at 16:36
@ Mercy , @ GTX OC and @J.W.Perry thank you all for your comments and hints. I read and I solved.(Sorry I computed :). ). Thank you so much. — Airbag, Oct 26 '13 at 18:10

score 1 · Answer 1 · answered Oct 26 '13 at 16:20

1

HINT

One definition of the exponential function is $$\operatorname{e}^u = \lim_{n\to\infty}\left( 1 + \frac{u}{n} \right)^n $$

In your case $u = -\tfrac{1}{2}x^2$. An appropraite substitution should help you continue.

answered Oct 26 '13 at 16:20

Fly by Night

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Definite integral: $\lim\limits_{n\to\infty}\int_{-\sqrt n}^{\sqrt n}\left(1-\frac{x^2}{2n}\right)^n \,\mathrm dx$

1 Answers1