\( a_n=\int_0^1x^{np}(x^p+1)^n\mathrm {dx} \) , unde \( p\in\mathbb N^* \) \( \ \Longrightarrow\ \lim_{n\to\infty}\ \frac {n}{2^n}\cdot a_n=\frac {2}{3p} \) .
Cazuri particulare :
\( p:=1\ \Longrightarrow\ \lim_{n\to\infty}\ \frac {n}{2^n}\cdot\sum_{k=0}^n\frac {C_n^k}{n+k+1}=\frac 23 \) ;
\( p:=2\ \Longrightarrow\ \lim_{n\to\infty}\ \frac {n}{2^n}\cdot\sum_{k=0}^n\frac {C_n^k}{2n+2k+1}=\frac 13 \) .
Un sir exprimat printr-o integrala definita
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Virgil Nicula
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Marius Mainea
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Propozitie: Fie a,b\( \in\mathbb{R} \) cu a<b f:[a,b]\( \rightarrow[0,1) \) o functie strict crescatoare si derivabila in b , atunci:
i) Daca f(b)<1 atunci \( \lim_{n \to\infty}n\cdot\int_a^b f^n(x)dx=0 \)
ii) daca f(b)=1 atunci \( \lim_{n\to\infty}n\int_a^bf^n(x)dx=\left\{ \begin{array}{rcl}+\infty & \mbox{daca} & f^\prime(b)=0\\\frac{1}{f^\prime(b)} & \mbox{daca} & f^\prime(b)\neq0 \end{array}\right. \)
i) Daca f(b)<1 atunci \( \lim_{n \to\infty}n\cdot\int_a^b f^n(x)dx=0 \)
ii) daca f(b)=1 atunci \( \lim_{n\to\infty}n\int_a^bf^n(x)dx=\left\{ \begin{array}{rcl}+\infty & \mbox{daca} & f^\prime(b)=0\\\frac{1}{f^\prime(b)} & \mbox{daca} & f^\prime(b)\neq0 \end{array}\right. \)