Mathematics

Problem

Let

k

be a real number and consider the integral

L(k)=\int_{0}^{\infty} \frac{x^{k}}{e^{x}-1}\,dx.

Throughout this problem assume

-1<k<\infty

so that the integrand is defined on

(0,\infty)

. (a) Identify all points and/or intervals where

L(k)

is improper and state why. (b) Determine all real values of

k

(-1,\infty)

for which

L(k)

converges. Your justification should analyze the behavior of the integrand near each improper point and employ suitable comparisons. (c) The function

\Gamma(s)=\int_{0}^{\infty} x^{s-1}e^{-x}\,dx

is known to converge for

s>0

. Explain how the convergence result from part (b) relates to the domain of

\Gamma

without computing any values.