Proving the Infinite Series: (xlna)^(n-1)/n!

[Kevin Huang]

Homework Statement

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

Homework Equations

Maclaurin series expansion

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.

 

[D H]

Note that

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

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