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• 14:07, 28 October 2023 Ghetie Gabriela Claudia talk contribs created page 28250 (Pagină nouă: _{'''<big>28250 (Codruț-Sorin Zmicală)</big>'''} ''Calculați'' ''<math>\lim_{n \to \infty}\sqrt[n]{\int_{0}^{1} (\sqrt{x}+x^n})^ndx</math>.'' '''Soluție:''' Fie <math>a_n=\int_{0}^{1} (\sqrt{x}+x^n)^ndx</math>, n<math>\in\Nu^*</math>. Cu binomul lui Newton avem <math>(\sqrt{x}+x^n)^n=\sum_{k=0}^n\binom{n}{k}x^\tfrac{(2n-1)k+n}{2}</math>, iar prin integrare pe [0,1] obținem <math>a_n=\sum_{k=0}^n\binom{n}{k}\cdot\frac{2}{(2n-1)k+n+2}</math>. Pentru orice <ma...) Tag: Visual edit