From an integral to a gamma to a series

In summary, the conversation discusses using Euler's Gamma and a proper substitution to prove the relation \int_0^1 1/x^x=\sum_{n=1}^\infty 1/n^n. The conversation suggests expanding x-x = e -x ln x as a series and then using a nice substitution, such as the exponential, to simplify the problem. The conversation concludes with the suggestion to do a small change of variables to write the integral as a gamma function.
  • #1
deathquasar
4
0

Homework Statement


Using Euler's Gamma and a proper substitution prove the relation above:
[itex]\int_0^1 1/x^x=\sum_{n=1}^\infty 1/n^n[/itex]


Homework Equations


How to resolve this XD?


The Attempt at a Solution

 
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  • #2
Maybe you should start by expanding x-x = e -x ln x as a series and then trying to think of a nice substitution.
 
  • #3
but I've to pass through the euler's gamma representation, and expanding this function is quite horrible using taylor. I'll try anyway with your idea!,Ty!
 
  • #4
Don't expand it into a full Taylor series, just the exponential. That should give you a clue what the substitution should be (remembering that you want a gamma function in the end).
 
  • #5
ok, I expanded just the exp, and now I've this, (with some transformations) and quite looks like what I want.

[itex]\sum_1^\infty \frac{1}{n!}\int_0^\infty e^{-y(n+1)}y^n[/itex]

but now?
 
  • #6
Now do a small change of variables so you can write the integral as a gamma function
 
  • #7
ok done :D thank you very much ^^
 

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is a fundamental operation in calculus and is used to find the total value of a function over a specific interval.

2. What is a gamma function?

A gamma function is a special type of mathematical function that is used to extend the concept of factorial to non-integer values. It is commonly denoted as Γ and is defined as the integral of the function xn-1e-x from 0 to infinity.

3. How is a series related to an integral?

A series is a sum of terms in a sequence, while an integral is a mathematical operation that finds the area under a curve. The relationship between the two is that the integral of a function can be approximated by a series of smaller, simpler functions.

4. Why is the gamma function important?

The gamma function has many important applications in mathematics and physics. It is used in the fields of probability, statistics, number theory, and quantum mechanics. It also has applications in solving differential equations and evaluating complex integrals.

5. How is the gamma function related to the factorial function?

The gamma function is an extension of the factorial function, which only applies to positive integers. The gamma function allows for the calculation of factorial values for non-integer values, making it a more versatile tool in mathematics.

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