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An integral with exponential tail

Calculus (Integrals, Series)
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Riemann
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An integral with exponential tail

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Post by Riemann » Mon Sep 19, 2016 10:51 am

Let $n \in \mathbb{N}$ and let us define the function $\displaystyle \mathcal{E}_{\tau, n} (x) = \int_{0}^{x} e^{-t} t^n \, {\rm d}t$ better known as lower incomplete gamma function. Prove that:

$$\bigintsss_{0}^{1}\left ( e^x - 1 - x - \frac{x^2}{2} - \cdots - \frac{x^n}{n} \right ) \, {\rm d}x = \frac{e}{n!} \mathcal{E}_{\tau, n} (1) - \frac{1}{n!(n+1)}$$
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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