Infinite series of (x/k)^k

[ileslie]

I made up this series for a class the other day. It is a sum for k = 1,2,3, ... inf
of the terms (x/k)^k. It gets big rather quickly but will always converge for x >=0.
Once k > x the terms get small very quickly. Here is my question. Does anyone recognize this series? I have not been able to find it discussed online or in texts in the library. I am curious if it represents some known function(s). If not, has some asymptotic analysis been done. The sum seems to approach sqrt(c*x)*exp(x/e) for large x (x>400), with c close to 2.3104048729. This was found with Matlab, but I've made no serious attempt at any analysis.

Thanks in advance.

[acheong87]

i don't know the complete answer, but as a hint, look up "tetration" (on wikipedia, for example). your series is essentially the sum of (x^k)/(k^k). i'm not sure that you'll find an answer easily, because i don't think this summation is very useful in application. although i've only taken basic analysis, i think that k^k does not make much dimensional (physical?) sense.

tetration is a very very scary operator. thats all i have to say.