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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?
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.
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 x^{n-1}e^{-x} from 0 to infinity.
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.
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.
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.