Answer to Question #142195 in Real Analysis for Amit

Question

f:[0,1]→R be Riemann integrable. Find lim┬(n→∞)⁡∫_0^1▒〖x^(n+1) f(x)dx〗.

Answer 1

Given $f:[0,1]\to \R$ is Riemann integrable function.

We need to find

$$\lim_{n\rightarrow \infty}\int_0^1x^{n+1}f(x)\,dx$$

Using Integration by parts we get,

$$\frac{f(x)x^{n+2}}{n+2}\bigg|_0^1-\frac{1}{n+2}\int_0^1f'(x)x^{n+2}\,dx\\ \implies \lim_{n\rightarrow \infty}\int_0^1x^{n+1}f(x)\,dx=0$$