Series expansion of exponential form

Question

What is the simplified results of $1+e^{j\theta}+e^{j2\theta}+\cdots+e^{jM\theta}$ as $M$ goes to infinity? First it looks like Taylore series expansion of the x to the power $2,3,\ldots$ but as the $e^{j\theta}$ is a complex number and $e$ by itself is a number bigger than $1$ I am not sure how to solve this.

Answer 1

Recall that by geometric series for $e^{j\theta}\neq 1$

$$1+e^{j\theta}+e^{j2\theta}+...+e^{jM\theta}=\frac{1-e^{j(M+1)\theta}}{1-e^{j\theta}}$$