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Does anyone know how to evaluate the infinite product

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

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$k=2 \to 1 - \frac{4}{k^2}=0$ – hhsaffar Dec 22 '13 at 23:49
Do you possible mean to start the index at $k = 3$? – user61527 Dec 22 '13 at 23:50
Modify the start. There is pleasant telescoping. – André Nicolas Dec 22 '13 at 23:50
Oops... I mistyped the question. I have now fixed it. – Geoff Smith Dec 22 '13 at 23:51

For $k \ge 3$:

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

This is equal to

$$\frac{1\cdot 5}{3 \cdot 3} \frac{2\cdot 6}{4 \cdot 4} \frac{3\cdot 7}{5 \cdot 5} \cdots$$

With cancellations: note that only $1\cdot 2$ survives in the numerator, and a single $3 \cdot 4$ survives in the denominator. Thus, the product is $1/6$. The front factor produces a $-3$, so the stated product is $-1/2$.

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Without the guesswork you could also go like this : \begin{align*} \prod_3^n (1-\frac{4}{k^2}) =&\ \prod_3^n \frac{k^2-4}{k^2}= \prod_3^n \frac{(k+2)(k-2)}{k^2}\\ =&\ \prod_3^n (k+2) \ \prod_3^n (k-2) \ \frac{1}{\prod_3^n k^2}\\ =&\ \frac{(n+2)!}{4!} \ (n-2)! \ \frac{1}{(n!/2)^2}\\ =&\ \frac{n!(n+1)(n+2)}{24} \ \frac{n!}{n(n-1)} \ \frac{4}{n!^2}= \frac{(n+1)(n+2)}{6 n (n-1)} \rightarrow 1/6 \end{align*} – A.G. Dec 23 '13 at 3:15
@A.G.: there's no guesswork - it's a matter of observing which numbers appear on top or bottom once and which ones twice (or not at all). That's it. But sure, your way does the job; I just tried to keep it as simple as possible. – Ron Gordon Dec 23 '13 at 5:44

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

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[Comment about how this is overkill] – Pedro Tamaroff Dec 23 '13 at 0:34
[Comment defending alternative methods of evaluation and the importance of complex analysis] – Pedro Tamaroff Dec 23 '13 at 0:34

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