Does anyone know how to evaluate the infinite product
$$
\left(1 – \frac{4}{1}\right) \prod_{k = 3}^{\infty} \left( 1 – \frac{4}{k^2} \right)
$$
[Math] Evaluating an Infinite Product
infinite-product
infinite-product
Does anyone know how to evaluate the infinite product
$$
\left(1 – \frac{4}{1}\right) \prod_{k = 3}^{\infty} \left( 1 – \frac{4}{k^2} \right)
$$
Best Answer
Here is an alternative proof that does not use telescoping. We have that $$\frac{\sin(\pi z)}{\pi z}=\prod_{k=1}^\infty \left(1-\frac{z^2}{k^2}\right).$$ This can be proven by using the Weierstrass product for the Gamma function combined with Euler's reflection formula. Dividing both sides by $1-\frac{z^2}{4}$, we see that $$\prod_{k\neq 2}\left(1-\frac{4}{k^2}\right)=\lim_{z\rightarrow 2} \frac{4\sin(\pi z)}{\pi z(2-z)(2+z)}.$$ Taylor expanding $\sin(\pi z)$ around $z=2$, we are able to conclude that $$\prod_{k\neq 2}\left(1-\frac{4}{k^2}\right)=-\frac{1}{2}.$$