One-liner mini-challenge [HP-71B]

[Valentin Albillo]

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Hi, Gerson:

(04-17-2015 01:46 PM)Gerson W. Barbosa Wrote:  Perhaps this is a more interesting problem:

\[\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{5}+\cdots+\sqrt{n}\]

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This 69-char HP-71B one-liner will probably do as well:

INPUTN@2/3*N^1.5+N^.5/2-0.207886224977+(1-1/80/N^2+1/384/N^4)/N^.5/24

which gives 12-digit precision for arguments as low as 7 or 8 and even 6-digit precision (save for a few ulps) for arguments as low as 1 ! (1,00003...).

The constant is Zeta(-1/2). Of course one or several final terms can be omitted if desired for a shorter expression with somewhat reduced accuracy.

Regards.
V.
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