Euler's Theorem

Euler's Theorem

We already saw a justification for Euler's equation

using differential equations back when we discussed complex numbers. However, a more conclusive proof is available to us now using power series.

Consider the power series for cosine and sine:

Now consider the power series for :

Example: Find a simplified form for

We can rewrite this sum using exponential functions

Each of the series in brackets is a geometric series and hence we have a simple way to find the sum.