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@robinhouston - an amusing fact: in 1995, in a paper on the arXiv, Kent Morrison said "we might conjecture" that

\[
\int_0^\infty \prod_{n=1}^\infty \cos(x/n)\,dx \stackrel{?}= \pi/4 \]

\[ \int_0^\infty \cos(2x) \prod_{n=1}^\infty \cos(x/n)\,dx \stackrel{?}= \pi/8 \]

Both these are false, of course! I read about this here:

https://mathoverflow.net/q/354932/2893