An integral with exponential tail

Calculus (Integrals, Series)
Post Reply
User avatar
Riemann
Posts: 176
Joined: Sat Nov 14, 2015 6:32 am
Location: Melbourne, Australia

An integral with exponential tail

#1

Post by Riemann »

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:

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

Create an account or sign in to join the discussion

You need to be a member in order to post a reply

Create an account

Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute

Register

Sign in

Who is online

Users browsing this forum: No registered users and 17 guests