1. ## Power Series

Show that the power series $\displaystyle f(z) = 1+z+\frac{z^2}{2!}+ \cdots = \sum_{n=1}^{\infty} \frac{z^n}{n!}$ is equal to $\displaystyle e^z$.

So suppose $\displaystyle w \in \mathbb{C}$. Then $\displaystyle f(z)f(w) =\sum_{n=1}^{\infty} \frac{z^n}{n!} \cdot \sum_{n=1}^{\infty} \frac{w^n}{n!} = f(z+w)$. Also $\displaystyle f(x) = 1+x+\frac{x^2}{2!}+ \cdots = \sum_{n=1}^{\infty} \frac{x^n}{n!}$. And so $\displaystyle f(x) = e^{x}$. And then we need to show that $\displaystyle f(iy) = \cos y + i \sin y$?

2. In other words we know that $\displaystyle \exp(wz) = \exp w + \exp z$. So I established that in the first part by showing that $\displaystyle f(z)$ follows that property. So $\displaystyle e^z = e^{x+iy} = e^{x}e^{iy}$. So I evaluated the real and imaginary parts of $\displaystyle f(z)$.