TIFR-2011-Maths-A-17

Question

The value of the infinite product

$\prod_{n=2}^{\infty} (1-\frac{1}{n^{2}})$

is 1.

Answer 1

$\left(1- \frac{1}{n^2} \right ) = \frac{n^2-1}{n^2} = \frac{(n-1)(n+1)}{n^2}$

$\prod_{n=2}^{\infty} \left(1- \frac{1}{n^2} \right ) = \prod_{n=2}^{\infty} \frac{(n-1)(n+1)}{n^2}$

$= \frac{1.3}{2.2}\times\frac{2.4}{3.3}\times\frac{3.5}{4.4}\times\frac{4.6}{5.5}\times...$

$= \frac{1}{2} \neq 1$

[for ending terms of product, $n\rightarrow \infty, \frac{1}{n}\rightarrow 0,\left(1-\frac{1}{n^2} \right )\rightarrow 1$]