Applications

Abstract

In this section, we derive the formula

$$\begin{array}{lll}{\int^1_0}\frac{dx}{x^x}&=\sum^\infty_{n=1}\frac{1}{n^n}\\&=\frac{1}{1^1}+\frac{1}{2^2}+\frac{1}{3^3}+\cdots .\end{array}$$

(5.1.1)

Along the way we meet Euler’s gamma function and the monotone convergence theorem, both of which play roles in subsequent sections.