Seemous 2009, Problema 1

[Alin Galatan]

a) Calculati limita \( \lim_{n\to\infty}\frac{(2n+1)!}{(n!)^2}\int_0^1(x(1-x))^nx^kdx \), unde \( k\in N \).

b) Calculati limita \( \lim_{n\to\infty}\frac{(2n+1)!}{(n!)^2}\int_0^1(x(1-x))^nf(x)dx \), unde f e o functie continua pe [0,1].

[Cezar Lupu]

Asa ca hint, e bine de stiut ca, daca \( I_{n}=\int_0^1x^{n}(1-x)^{n}dx \), atunci are loc formula \( I_{n}=\frac{(n!)^{2}}{(2n+1)!} \).