Limit $\lim_{n \to \infty} \int_{0}^{1} n x^n f(x) dx $

Asked Sep 06 '20 at 08:51

Active Sep 06 '20 at 10:20

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I am unable to solve this limit of integral asked in a quiz and so looking for help here.

Question : Let $f:[0,1]\to \mathbb{R}$ be a continuous function. Then what is the following limit: $\lim_{n \to \infty} \int_{0}^{1} n x^n f(x) dx $. Give the reasons.

If $n {x}^n f(x) $ was uniformly convegent then $\lim_{n \to \infty }$ could be taken inside the integral but there is not enough information regarding the behaviour of $f(x)$ to check for uniform convergence.

So, Kindly tell how should I proceed.

Thanks!!

edited Sep 06 '20 at 10:20

jimjim

9,855

asked Sep 06 '20 at 08:51

Use the substitution $u = x^{n+1}$ – Ninad Munshi Sep 06 '20 at 08:53
Also: https://math.stackexchange.com/q/864442/42969 and https://math.stackexchange.com/q/128823/42969 – all found with Approach0 – Martin R Sep 06 '20 at 08:59

Limit $\lim_{n \to \infty} \int_{0}^{1} n x^n f(x) dx $

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