I'm tasked with finding the following limit:

$$\lim_{n\to\infty} \int_0^1 \frac{(nx)^2}{(1+x^2)^n}dx$$

I think the right way to do this is to swap the integral and the limit to get the answer $0$, but I'm having trouble justifying this operation using e.g. monotone convergence or dominated convergence. Alternatively, I could find an upper bound on the integral which converges to zero.

  • $\begingroup$ Are you sure it's actually $0$? From a computation, it looks like it actually increases as a function of $n$, and grows at least more quickly than $\log n$. $\endgroup$ – anomaly Sep 25 '17 at 22:27

Make the change of variables $x=y/\sqrt n$ to see the expression equals

$$\tag 1 \sqrt n \int_0^{\sqrt n} \frac{y^2}{(1+y^2/n)^n}\,dy.$$

Now $(1+u/n)^n \le e^u$ for $u\ge 0.$ Therefore $(1)$ is at least

$$\sqrt n \int_0^{\sqrt n} y^2e^{-y^2}\,dy.$$

The limit of this is $\infty,$ hence the limit in question is $\infty.$


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