My personal favorite relation between π and e 
Is there any other mathematician whose work is instantly recognizable?pic.twitter.com/WMliDN17Vs
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I don't see any obvious argument as to whether this sum is rational or irrational.
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It looks like the standard proof of the irrationality of e works here too: multiply the series by a large enough integer, wlog 3*5*…*(2k-1). It then = (integer) + 1/(2k+1) + 1/(2k+3) + …. The sum is bounded by sum_n 1/(2k+1)^n = 1/(1-1/(2k+1)) - 1 < 1, so never integer itself
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