Limit of integral of a sequence of functions (check)

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.

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}

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