How to use dominated convergence theorem?

Question

How to use dominated convergence theorem to compute $$lim_{n\rightarrow \infty}\int_0^1\frac{1+nx^2}{(1+x^2)^n}$$

So far I have only done $\frac{1+nx^2}{(1+x^2)^n}\le\frac{1+nx^2}{(1+x^2)}$, I don't know what to do next with $1+nx^2$. Can anyone help me with that? Thanks so much!

Answer 1

By Bernoulli's inequality, for $n \geq 0$ and $x \geq -1$, we have

$$ (1+x)^n \geq 1+nx $$

Replacing $x$ by $x^2$ you get a dominator $1$.

Answer 2

You don't necessarily want to throw away the power $n$ in the denominator.

Here is the idea: since $1 + x^2 \geq 1$, $(1 + x^2)^n$ grows geometrically with $n$ for fixed $x\neq 0$, and therefore much faster than $nx^2$ which only grows linearly with $n$. Therefore as a crude estimate $$ \frac{1 + nx^2}{(1+x^2)^n} = \frac{1}{(1+x^2)^n} + \frac{nx^2}{(1+x^2)^n} \leq 1 + \epsilon $$ for large $n$ and $x\neq 0$. In fact for $x \neq 0$ you can do even better (though not $x=0$, but there the limit is obvious); this suggests what the limit will be.