Comparing a series to an integral

Abstract

We consider the difference between the definite integral ${\int }_0^{\infty }{u}^x{e}^{-u}du$, where $x$ is a real parameter, and the approximating sum ${\sum }_{k=1}^{\infty }{k}^x{e}^{-k}$. We use properties of Bernoulli numbers to show that this difference is unbounded and has infinitely many zeros. We also conjecture that the sign of the difference at any positive integer $n$ is determined by the sign of $cos\left(\left(n+1\right)arctan\left(2\pi \right)\right)$.