Suggesting closed-form representations of mathematical constants by means of experimental mathematics?

I can't resist giving a couple of examples of (contrived) numerical coincidences (both are from Experimentation in Mathematics by J.M. Borwein, D.H. Bailey and R. Girgensohn).

Example 1.

$$\int_{0}^{\infty}\cos(2x)\prod_{k=1}^{\infty}\cos(x/k)\ dx=\frac{\pi}{8}-\epsilon,$$ where $0<\epsilon<10^{-41}.$

Example 2.

$$\sum\limits_{k=1}^{\infty}e^{-(k/10)^2}\approx5\sqrt\pi-\frac{1}{2}=8.362269254527580...$$

Well, they agree through 427 (four hundred twenty seven) digits yet they are not equal.

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A moral. Make sure you understand the context and use your inverse symbolic calculator with caution.

Have you tried the Inverse Symbolic Calculator?

The rational constant for an Apery like series for $$\zeta(4)$$ was found experimentally, using continued fractions.