prove-that-n-2-1-1-n-2-1-2-

Question Number 90561 by Tony Lin last updated on 24/Apr/20

p r o v e t h a t

\underset{n = 2}{\prod^{\infty}} (1 - \frac{1}{n^{2}}) = \frac{1}{2}

Commented by mathmax by abdo last updated on 24/Apr/20

l e t S_{n} = \prod_{k = 2}^{n} (1 - \frac{1}{k^{2}}) \Rightarrow S_{n} = \prod_{k = 2}^{n} \frac{k^{2} - 1}{k^{2}}

= \prod_{k = 2}^{n} \frac{k - 1}{k} \times \frac{k + 1}{k} = \prod_{k = 2}^{n} \frac{k - 1}{k} \times \prod_{k = 2}^{n} \frac{k + 1}{k}

= \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \dots . \frac{n - 1}{n} \times \frac{3}{2} \times \frac{4}{3} \times \dots . \times \frac{n}{n - 1} \times \frac{n + 1}{n}

= \frac{n + 1}{2 n} \Rightarrow {l i m}_{n \to + \infty} S_{n} = \frac{1}{2}

Commented by Tony Lin last updated on 25/Apr/20

t h a n k s s i r