Ideas To Show this statement: $\prod_2^{2n+1}(1-k^{-2})=(n+1)/(2n+1)$

Question

I would like to know an idea on how to show this:

$\prod_{k=2}^{2n+1}$ $(1-{1\over k^2})$ = ${n+1\over 2n+1}$ $\forall$ $n\ge$ $2$.

I already checked for $2$ and tried it by induction but I didn't succeed.

Answer 1

Two hints to start you off:

1) You do not need induction. The statement can be proved purely by algebraic manipulation of the given formula.

2) Write $$1 - \frac{1}{k^2} = \frac{k^2-1}{k^2}.$$ This can be put into the product easily. Think about factoring the terms here, and factorials in the product. Many terms will cancel.