Question

Ma poate ajuta cineva la exercitiul 2, b si c? La b presupun ca trebuie sa gasesc o formula de recurenta dar nu prea imi iese

Answer 1

b)

$I_{n+1} \leq I_{n} <=> I_{n+1} - I_{n} \leq 0$

$I_{n+1} - I_{n} = \int\limits^1_0 {xe^{-(n+1)x^{2}} - xe^{-nx^{2}} } \, dx =$
$\int\limits^1_0 {xe^{-nx^{2}-x^{2}} - xe^{-nx^{2}} } dx \\ \\ = \int\limits^1_0 {xe^{-nx^{2}}*e^{-x^{2}} - xe^{-nx^{2}} } dx$

$= \int\limits^1_0 {xe^{-nx^{2}}(e^{-x^{2}} - 1) }\ dx$

$Pentru \ x \ apartine \ [0,1] \ si \ 'n' \ natural, avem$

$e^{-nx^{2}} > 0 \ si \ e^{-x^2} -1 \leq 0 \\ \\ => I_{n+1} - I_{n} \leq 0$

c)

$I_{n}= \int\limits^1_0 { \, xe^{-nx^{2}}} dx$

$= - \frac{1}{2n} \int\limits^1_0 {(e^{-nx^{2}})' dx$
$=- \frac{1}{2n}e^{-nx^{2}} \ de \ la \ 0 \ la \ 1$
$= \frac{1}{2n} (1- \frac{1}{e^n} )$