n! / n^n

[FRANCISCO SALINAS]

Ross wrote:

>I'm interested in the sum of n!/n^n, i.e. Sum[n=1 to infinity, n!/n^n].  I 
>*believe* this converges to very roughly 1.879853...

1.87985386217525853348630614507096003881987340048928990482961766912229638666121421136176501973891235323968423504123314722674855792558795100996638203202230520427799290955212115586554499729555606646791467............

>If so, is it at all interesting...?

I think so. By itself is interesting, and it's also a standard example for 
series convergence tests.

>And if it is, perhaps it's decimal expansion could be considered for 
>addition to the OEIS ala pi, e etc...?

Undoubtelly, yes. There's an entry for it in Plouffe's Inverter

>From Stirling it should approximate to (sqrt(2*pi*n))/e^n.  I realize there 
>is probably nothing of interest here, but am just curious.

If you are looking for a closed form, I'd be surprised if there exists one. 
Although we can obtain a beatiful integral representation:

1 + int(1/(exp(x)-x)^2,x=0..infinity)

The case is very similar to that of Sum[n=1 to infinity, 1/n^n]= 1 + 1/2^2 + 
1/3^3 + ... ( see A073009 or A083649 and 
http://mathworld.wolfram.com/PowerTower.html) for which is only known a 
'non-elementary' integral representation: int(1/x^x,x=0..1);

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