An interesting power series

Franklin T. Adams-Watters franktaw at
Sun Sep 25 21:46:41 CEST 2005

I'm not sure this really belongs on this mailing list, but I don't know where else to put it.

I've been looking at the function sum_{n=0}^{infinity} x^n/n^n.  Of particular interest is the behavior of the function for negative values of x.

Trivially, f(0) = 1.  f(1) =
= A073009 + 1.  f(-1) =
= 1 - A083648.

For small negative values of x, f is decreasing, with a root at x =
It reaches a minimum at x =
where the value is
after which it gradually heads back towards zero.  None of these three numbers are in the OEIS.

This function is closely related to the so-called "Sophomore's Dream" <>.  Generalizing the formulas there, we have:

Integral_0^1 t^(x*t) dt = -1/x * Sum_{n=1}^{infinity} (-x)^n/n^n = (1 - f(-x))/x;

or, inverting,

f(x) = 1 + x * Integral_0^1 t^(-x*t) dt.

Evaluating numerically, it appears that as x -> -infinity, f(x) acts like -1/log(|x|); possibly -1/log(|x|) + O(log(log(|x|))/log(|x|)^2), although the latter term is far from certain.  This is equivalent to

limit x->infinity Integral_0^1 t^(x*t) dt = 1/x + 1/(x*log(x)) + O(1/(x*log(x)^2)) [if the error term is correct].

I have no idea how to evaluate this limit to see if this is correct.

It appears that as x -> infinity, f(x) ~ sqrt(x)*exp(x/e).  This is what one would expect from Stirling's approximation, but again I am unable to prove it.
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067

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