Today we shall show the relationship between complex numbers and Maclaurin’s Series and prove the Euler’s Formula, i.e.

$latex e^{i \theta} = \text{cos}\theta + i \text{sin}\theta$.

Recall we have the following from Maclaurin’s Series.

$latex e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots$

$latex \text{sin}x = x – \frac{x^3}{3!} + \ldots$

$latex \text{cos}x = 1 – \frac{x^2}{2!} + \ldots$

Let us start our proof from the LHS.

$latex e^{i \theta}$

$latex = 1 + (i \theta) + \frac{(i \theta) ^2}{2!} + \frac{(i \theta) ^3}{3!} + \ldots$

$latex = 1 + i \theta – \frac{\theta ^2}{2!} – i \frac{\theta ^3}{3!} + \ldots$

$latex = 1 – \frac{\theta ^2}{2!} + i \theta – i \frac{\theta ^3}{3!} + \ldots$

$latex = 1 – \frac{\theta ^2}{2!} + \ldots + i \big( \theta – \frac{\theta ^3}{3!} + \ldots \big)$

$latex = \text{cos}\theta + i\text{sin}\theta$

∑23ƒ1

√11–21>3≤12