here's a good explanation ...
Fermat's Last Theorem: Euler's Formula
The general idea behind Maclaurin Series is to show you how to write functions like , and as an infinite power series in . (The Taylor series is similar, but is a bit more complicated. In fact, the Maclaurin is a special case of the Taylor series.)
If you don't know what I mean by an infinite power series, you're probably best to start by finding out about Infinite Geometric Series first. This will help you to understand how an infinite series can have a finite (and useful!) sum.
It's too complicated to show you here how the Maclaurin series is derived and used, but if you look at the .pdf file at http://mathinsite.bmth.ac.uk/pdf/macseries_theory.pdf you'll find a thorough explanation.
You need to look in particular at:
- Page 4, where you'll find the formula for the Maclaurin Series itself, and how it gives you the series for and
- Page 6, where you'll find the series for
To use these to prove Euler's formula, is very straightforward:
- Multiply both sides of the series by , noting that , etc.
- Add the result to the series for , arranging the terms in ascending powers of .
- In the series expansion of , replace by . Note this time that , etc.
- Compare the result with what you found for . You should find they're equal.
There's a proof in the Wikipedia article at Euler's formula - Wikipedia, the free encyclopedia.
Hope that helps.