Question

Calculati:
$\lim _{x\to \infty }\left(\left(1-\frac{1}{2^2}\right)\cdot \left(1-\frac{1}{3^2}\right)\cdot ...\cdot \left(1-\frac{1}{n^2}\right)\right)$

Answer 1

Salut,

Limita din enunț se referă la un produs, cu termenul general, cu k de la 2 la n:

$1-\dfrac{1}{k^2}=\dfrac{k^2-1}{k^2}=\dfrac{k-1}k\cdot\dfrac{k+1}k.\ Produsul\ devine:\\\\\dfrac{1}2\cdot\dfrac{3}2\cdot\dfrac{2}3\cdot\dfrac{4}3\cdot\ldots\cdot\dfrac{n-2}{n-1}\cdot\dfrac{n}{n-1}\cdot\dfrac{n-1}{n}\cdot\dfrac{n+1}{n}=\dfrac{1}2\cdot\dfrac{n+1}n\to\dfrac{1}2.$

Limita este deci 1/2.

Simplu, nu ? :-).

Green eyes.