Show that exponential of a Pauli matrix $A$ can be simplified as follows:
$$ \exp(i A \theta) = \cos(\theta) I + i \sin(\theta) A $$
Let's expand the Taylor series for $\exp(iA\theta)$:
$$ \exp(iA\theta) = \sum_{n = 0}^{\infty} \frac{(iA\theta)^n}{n!} = 1 + \frac{iA\theta}{1!} + \frac{(-1)A^2\theta^2}{2!} $$ $$ + \frac{(-i)A^3\theta^3}{3!} + \frac{A^4\theta^4}{4!} + \frac{iA^5\theta^5}{5!} + \cdots $$
How can we simplify this infinite sum?