Suma Riemann de integrale improprii

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Cezar Lupu
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Suma Riemann de integrale improprii

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Consideram \( f: (0,1) \to\mathbb{R} \) o functie monotona astfel incat integrala improprie \( \int_0^1f(x)dx \) sa existe. Sa se arate ca

\( \lim_{n\to\infty}\frac{1}{n}\left(f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\ldots+f\left(\frac{n-1}{n}\right)\right)=\int_0^1f(x)dx \).
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second orders half a beer. The third, a quarter of a beer. The bartender says “You’re all idiots”, and pours two beers.
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