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]\\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\\\\\n\\implies \\lim_{n\\rightarrow \\infty}\\int_0^1x^{n+1}f(x)\\,dx=0"


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