Limita de sir integral cu radicali

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Cezar Lupu
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Limita de sir integral cu radicali

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Sa se arate ca

\( \lim_{n\to\infty} n^{2}\left(\int_0^1\sqrt[n]{1+x^{n}}dx-1\right)=\frac{\pi^2}{12}. \)

Vojtech Jarnik, 2002, Categoria II
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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