I am doing this exercises, but I am stuck. I have to evaluate this limit $$ \lim_{n\to \infty}\int_{\mathbb{R}} f_n(x)dx $$ where $f_n(x)=(1+\frac{2x}{n})^ne^{-x^2}$.

I have seen that $f_n(x)\to e^{2x-x^2}$, so my idea is to use the dominated convergence theorem. The problem is that I am not able to find a function $g(x)$ such that $|f_n(x)|\leq g(x)$ $\forall x\in \mathbb{R}$ and $g$ is integrable.

Could you give me some advices on which $g$ to use? Thanks in advance!

  • 1
    $\begingroup$ why can't you just use $g(x) = e^{2x-x^2}$? $\endgroup$ – mathworker21 Jul 14 at 12:11
  • $\begingroup$ yes, you are right. $\endgroup$ – Alessandro Pecile Jul 14 at 12:32

$|f_n(x)| \leq e^{2|x|-x^{2}}$ which is integrable. [$(1+\frac a n)^{n} \leq e^{a}$ for all $a >0$].


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