Question

sa se calculeze produsul urmator
(1-1/4)(1-1/9)···(1-1/n²), n≥2

Answer 1

Dupa ce aducem la acelasi numitor in paranteze, folosim la fiecare numarator fomula:

a²-b²=(a-b)(a+b) si obtinem:

$...=\dfrac{(2-1)(2+1)}{4}\cdot\dfrac{(3-1)(3+1)}{9}\cdot\ldots\cdot\dfrac{(n-1)(n+1)}{n^2}=$

$=\dfrac{1\cdot3}{2^2}\cdot\dfrac{2\cdot4}{3^2}\cdot\dfrac{3\cdot5}{4^2}\cdot\ldots\cdot\dfrac{(n-1)(n+1)}{n^2}=\dfrac{n+1}{2n}$