Answer to Question #142195 in Real Analysis for Amit

Question #142195
f:[0,1]→R be Riemann integrable. Find lim┬(n→∞)⁡∫_0^1▒〖x^(n+1) f(x)dx〗.
1
Expert's answer
2020-11-04T15:43:35-0500

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

We need to find


limn01xn+1f(x)dx\lim_{n\rightarrow \infty}\int_0^1x^{n+1}f(x)\,dx

Using Integration by parts we get,


f(x)xn+2n+2011n+201f(x)xn+2dx    limn01xn+1f(x)dx=0\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


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