Calculate the limit $$\lim_{n\to\infty} \int_0^n\left(1+\frac xn\right)^ne^{-2x}\,dx$$

In class we've seen two results, the dominated and monotone convergence theorems.

For both we need that $\lim_{n\to\infty}f_n$ exists almost everywhere. In this case, we have $\lim_{n\to\infty}f_n(x)=e^xe^{-2x}=e^{-x}$.

But I haven't been able to find any integrable function $g$, such that $|f_n(x)|\le g(x)$ and use the dominated convergence theorem, or a way to see if $f_n$ is non-decreasing and use the monotone convergence theorem.

I'm now also thinking if it's possible or useful to do that, since there's an $n$ also in the integration domain.

  • 1
    $\begingroup$ Do you know that $$\left(1+\frac x n\right) ^n \leq e^x$$? And multiply your integral with the indicator function on $[0,n)$ then you can integrate from 0 to infinity without any trouble. Can you finish after this? $\endgroup$
    – Shashi
    Jan 29 '18 at 13:37
  • $\begingroup$ Yes... It was so obvious and I couldn't see it.... Thank you! $\endgroup$
    – Raquel Gil
    Jan 29 '18 at 14:03

Well $(1+ \frac{x}{n})^n$ is monotonically increasing and so is $(1+ \frac{x}{n})^ne^{-2x}1_{[0,n]}$


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