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

  • Context: Graduate 
  • Thread starter Thread starter Jakim
  • Start date Start date
  • Tags Tags
    Sum
Click For Summary

Discussion Overview

The discussion revolves around the evaluation of the definite integral of the function x^x from 0 to 1, specifically focusing on the expression for the resulting sum and its potential representation in terms of the Meijer G-function.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant presents the integral of x^x from 0 to 1 and claims it can be expressed as a sum: \(\int_0^1 x^x = \sum_{n=1}^{\infty} \frac{(-n)^{1-n}}{n}\).
  • Another participant notes that the sum is known as the Sophomore's dream, suggesting it has established recognition.
  • A later reply indicates that the first participant was unaware of the name and expresses interest in further reading about it.
  • Additional references to a paper titled "Sophomore's Dream Function" are provided, which may contain relevant information.

Areas of Agreement / Disagreement

Participants generally acknowledge the existence of the sum known as the Sophomore's dream, but there is no consensus on its expression in terms of the Meijer G-function or other forms.

Contextual Notes

There may be limitations regarding the assumptions made in the integral evaluation and the dependence on specific definitions related to the G-function.

Jakim
Messages
4
Reaction score
0
Hi. I tried to evaluate a definite integral of [itex]x^x[/itex] from [itex]0[/itex] to [itex]1[/itex] and I have reached following sum:

[tex]\int_0^1 x^x = \sum_{n=1}^{\infty} \frac{(-n)^{1-n}}{n}[/tex]

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

Thanks in advance.
 
Physics news on Phys.org
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.
 
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.
 
Hi, I've downloaded it even :). It was an answer to some of my questions. Thanks.
 

Similar threads

Graduate Summing over continuum and uncountable numerocities
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Express the limit as a definite integral
  • · Replies 16 ·
Replies
16
Views
4K
Graduate Multiplying divergent integrals using Hardy fields approach
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Solving a power series
  • · Replies 3 ·
Replies
3
Views
4K
Graduate How to sum an infinite convergent series that has a term from the end
  • · Replies 15 ·
Replies
15
Views
3K
Undergrad What is the difference between these two power series theorems?
  • · Replies 5 ·
Replies
5
Views
3K
High School I don't recognize this limit of Riemann sum
  • · Replies 2 ·
Replies
2
Views
2K
High School How to merge the sum and ##x^n##?
  • · Replies 10 ·
Replies
10
Views
1K
Undergrad On convolution theorem of Laplace transform: Schiff
  • · Replies 4 ·
Replies
4
Views
3K
MHB Evaluate the surface integral $\iint\limits_{\sum}f\cdot d\sigma$
  • · Replies 1 ·
Replies
1
Views
3K