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

\begin{eqnarray*}
&=&\prod_{k=2}^{+\infty}\left(1-\frac1{k^2}\right)\\
&=&\prod_{k=2}^{+\infty}\frac{(k-1)(k+1)}{k^2}\\
&=&\frac{1\cdot3}{2\cdot 2}\cdot\frac{2\cdot4}{3\cdot3}\cdot\frac{3\cdot5}{4\cdot 4}\cdot\ldots\\
&=&\lim_{k\to+\infty}\frac{k+1}{2\cdot k}\\
&=&\frac12
\end{eqnarray*}
