<s>[INST] Compute $\prod_{k=5}^{\infty}\frac{k^4-17k^2+16}{k^4-8k^2+16}$ Compute the product: $$\prod_{k=5}^{\infty}\frac{k^4-17k^2+16}{k^4-8k^2+16}$$
I was able to factor in the following manner:
$$ \frac{k^4-17k^2+16}{k^4-8k^2+16}=\frac{(k-1)(k+1)(k-4)(k+4)}{(k-2)^2(k+2)^2}$$
but what should I do now?
 [/INST] Take the partial products up to $N$ and use the fact that $$\prod_{k=5}^N \frac{k-1}{k-2} = \frac{4}{3} \cdot \frac{5}{4} \cdot ... \cdot \frac{N-1}{N-2} = \frac{N-1}{3},$$ and similarly $$\prod_{k=5}^N \frac{k+1}{k+2} = \frac{6}{N+2}, \quad \prod_{k=5}^N \frac{k-4}{k-2} = \frac{2}{(N-2)(N-3)}, \quad \prod_{k=5}^N \frac{k+4}{k+2} = \frac{(N+4)(N+3)}{7 \cdot 8}.$$ So $$\prod_{k=5}^N \frac{k^4 - 17k^2 + 16}{k^4 - 8k^2 + 16} = \frac{(N-1)(N+3)(N+4)}{14 (N-3)(N-2)(N+2)}.$$
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