Testwiki:Reference desk/Archives/Mathematics/2019 February 27

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February 27

Fully faithful endofunctors of Set

Does there exist a fully faithful endofunctor F of the category of sets that is not an equivalence? GeoffreyT2000 (talk) 17:06, 27 February 2019 (UTC)

I don't think so? This is off the cuff, and I haven't taken the time to check it, but: Take a one-element set x. F must send it to a one-element set y. Now for any set z, by matching morphisms from x to z with morphisms from y to F(z), you should be able to understand the action of F on z as a bijection, which should then let you put together an inverse.--2404:2000:2000:5:0:0:0:C1 (talk) 22:55, 28 February 2019 (UTC)

Series

$S=\sum _{n=1}^{\mathrm{\infty }}{x}^{\ln (n)}$ How I calculate value of x.?!! If S was known. — Preceding unsigned comment added by 151.236.179.162 (talk) 18:32, 27 February 2019 (UTC)

Pretty sure the RHS is ζ(-log x) where ζ is the Riemann zeta function. I don't know if there are ways to compute ζ^-1 that are an improvement over standard numerical root-finding algorithms. For values of S like π²/6, which is known to be ζ(2), x would be easy to find. --RDBury (talk) 21:09, 27 February 2019 (UTC)