# Proving an Infinite Series

1. Jun 6, 2010

### Kevin Huang

1. The problem statement, all variables and given/known data
Given an infinite series that follows the form [(xlna)^(n-1)]/n!
n takes on integers from 0 onwards
x all real numbers
a all positive real numbers

2. Relevant equations
Maclaurin series expansion

3. The attempt at a solution
In which for the e^x series expansion plug in xlna into the x from e^x to obtain a^x which is the answer to the infinite summation. However, are there any other proofs besides using Maclaurin? Thanks.

2. Jun 6, 2010

### D H

Staff Emeritus
Note that

$$e^{ax} = \sum_{n=0}^{\infty} \frac {(ax)^n}{n!}$$

In other words the nth term of this series is $(ax)^n/n!$. You have a different series. The nth term of your series is $(ax)^{(n-1)}/n!$.