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

I was studying for some quizzes when wild question appears. It looks like this:

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

My work

I think it's a repeated multiplication of the expression $1-\frac{1}{k^2}$. It looks like this:

$$\prod_{k=2}^{+\infty}\left(1-\frac{1}{k^2}\right) = \left(1-\frac{1}{(2)^2}\right)\left(1-\frac{1}{(3)^2}\right)\left(1-\frac{1}{(3)^2}\right)\left(1-\frac{1}{(4)^2}\right).....$$

I barely had any experience evaluating these new summation...How do evaluate $\prod_{k=2}^{+\infty}\left(1-\frac{1}{k^2}\right)$?

• Try to find out what happens if $k=6$ (a small value of $k$). Much of the staff you have is canceled. – Konstantinos Gaitanas Sep 25 '17 at 8:43
• @KonstantinosGaitanas What do you mean "Much of the staff you have is canceled"? additional information is lost when I do the operation above? – Palautot Ka Sep 25 '17 at 8:52
• Look at the numerators and the denominators in Gerhard's answer below. – Konstantinos Gaitanas Sep 25 '17 at 8:53
• @KonstantinosGaitanas What topic in math do I learn this truth? because I just found this question during our review for some licensure exams.... – Palautot Ka Sep 25 '17 at 9:27