<s>[INST] Evaluating an Infinite Product Does anyone know how to evaluate the infinite product
$$
\left(1 - \frac{4}{1}\right) \prod_{k = 3}^{\infty} \left( 1 - \frac{4}{k^2} \right)
$$
 [/INST] 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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