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Take the natural logarithm of your product and you get a sum which can be evaluated:

$$\ln\left( \prod_{x=1}^k \frac{1}{x^4} \right) = \sum_{x=1}^k \ln\left(\frac{1}{x^4}\right)$$

... now take the limit as $k \to \infty$:

sage: sum(ln(1/x^4), x, 1, oo)
-Infinity
sage: e^sum(ln(1/x^4), x, 1, oo)
0