28250

_{28250 (Codruț-Sorin Zmicală)}

Calculați

$\lim _{n\to \infty }{\sqrt[{n}]{\int _{0}^{1}({\sqrt {x}}+x^{n}}})^{n}dx$.

Soluție:

Fie $a_{n}=\int _{0}^{1}({\sqrt {x}}+x^{n})^{n}dx$, n$\in \mathrm {N} ^{*}$. Cu binomul lui Newton avem $({\sqrt {x}}+x^{n})^{n}=\sum _{k=0}^{n}{\binom {n}{k}}x^{\tfrac {(2n-1)k+n}{2}}$, iar prin integrare pe [0,1] obținem $a_{n}=\sum _{k=0}^{n}{\binom {n}{k}}\cdot {\frac {2}{(2n-1)k+n+2}}$.

Pentru orice Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\in\{0,1,...,n\}} avem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom{n}{k}\cdot\frac{1}{n^2+1}\leqslant\binom{n}{k}\cdot\frac{2}{(2n-1)k+n+2}\leqslant\binom{n}{k}} , iar prin însumarea acestor inegalități obținem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2^n}{n^2+1}\leqslant a_n\leqslant 2^n} .

Rezultă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2}{\sqrt[n]{n^2+1}}\leqslant{\sqrt[n]{a_n}}\leqslant2} , pentru orice Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\geqslant2} . Cum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty}\sqrt[n]{n^2+1}=1} , din teorema cleștelui obținem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty}\sqrt[n]{a_n}=2} .