#### Vol. 7, No. 1, 2014

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Comparing a series to an integral

### Leon Siegel

Vol. 7 (2014), No. 1, 57–65
##### Abstract

We consider the difference between the definite integral ${\int }_{0}^{\infty }{u}^{x}{e}^{-u}\phantom{\rule{0.3em}{0ex}}du$, 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 $cos\left(\left(n+1\right)arctan\left(2\pi \right)\right)$.

##### Keywords
polylogarithms, gamma function, Bernoulli numbers
Primary: 33B15