What is the expression for the non-trivial sum of x^x from 0 to 1?

[Jakim]

Hi. I tried to evaluate a definite integral of x^x from 0 to 1 and I have reached following sum:

\int_0^1 x^x = \sum_{n=1}^{\infty} \frac{(-n)^{1-n}}{n}

Is there an expression of this sum, for example in terms of Meijer G-function? I tried to find x^x as G-function form to integrate it but unsuccessful.

Thanks in advance.

 

[Citan Uzuki]

I'm not sure whether it's related to the G-function or not, but the sum itself is well-known, and is called the Sophomore's dream.

 

[Jakim]

Thanks, I didn't know it has a name; I will look for articles about it. Anyway I have reached the sum the same way as it is shown in your link.

 

[JJacquelin]

Hi!

Some references in the paper :
"Sophomore's Dream Function"
http://www.scribd.com/JJacquelin/documents

 

[Jakim]

Hi, I've downloaded it even :). It was an answer to some of my questions. Thanks.