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 Post subject: Limit of an integralPosted: Sun Aug 06, 2017 6:43 am

Joined: Sat Nov 07, 2015 6:12 pm
Posts: 828
Location: Larisa
The following exercise is just an alternative of IMC 2017/2/1 problem. It is quite easy but it's not a bad idea to have it here as well.

Given the continuous function $f:[0, +\infty) \rightarrow \mathbb{R}$ such that $\lim \limits_{x \rightarrow +\infty} x^2 f(x) = 1$ prove that

$$\lim_{n \rightarrow +\infty} \int_0^1 f(n x)\, {\rm d}x =0$$

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 Post subject: Re: Limit of an integralPosted: Sat Sep 09, 2017 8:17 am

Joined: Sat Nov 14, 2015 6:32 am
Posts: 137
Location: Melbourne, Australia
Since $\lim \limits_{x \rightarrow +\infty} f(x) =0$ the result follows immediately by making the change of variables $u=nx$ .

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$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$

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