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Let $f_n:(0,1)\rightarrow \mathbb{R}$ be given by $$f_n(x)=n\,x^n.$$

I want to compute

$$lim_{n \rightarrow \infty}\int^{1}_{0}f_n(x)dx.$$

I have the following:

\begin{align} lim_{n \rightarrow \infty}\int^{1}_{0}f_n(x)dx &= lim_{n\rightarrow \infty}\int_0^1 n\,x^n dx\\ &=lim_{n\rightarrow \infty}\left[\frac{n\,x^{n+1}}{n+1}\right]_{x=0}^{x=1}\\ &=lim_{n\rightarrow \infty}\left[\frac{n}{n+1}\right]\\ &=1 \end{align}

I just want to confirm this is correct.

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Seems fine, just minor edit to remove $x=1$ and $x=0$ after you substituted them.

\begin{align} &\lim_{n\rightarrow \infty}\left[\frac{n\,x^{n+1}}{n+1}\right]_{x=0}^{x=1}\\ &=\lim_{n\rightarrow \infty}\left[\frac{n}{n+1}\right]\\ &=1 \end{align}

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  • $\begingroup$ Actually why can’t dominated convergence theorem be applied here? $\endgroup$ – Szeto Nov 4 '18 at 5:07
  • $\begingroup$ if the computation is correct, it means $f_n$ can't be dominated. Especially due to value close to $1$. $\endgroup$ – Siong Thye Goh Nov 4 '18 at 5:20

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