TITLE: Find sum of $1 + \cos \theta + \frac{1}{2!}\cos 2\theta + \cdots$
QUESTION [8 upvotes]: Find the sum of following series:
$$1 + \cos \theta + \frac{1}{2!}\cos 2\theta + \cdots$$
where $\theta \in \mathbb R$.
My attempt: I need hint to start.

REPLY [3 votes]: Hint to start
$$
1+\cos \theta + \frac{1}{2!}\cos(2\theta)+\cdots = \mathcal{Re}\sum_{k=0}^n\frac{\mathrm{e}^{ik\theta}}{k!} = \mathcal{Re}\sum_{k=0}^n\frac{\left(\mathrm{e}^{i\theta}\right)^k}{k!}
$$
Also remember that in the $\lim_{n\to\infty}$ we have
$$
\lim_{n\to\infty}\sum_{k=0}^n\frac{x^k}{k!}=\mathrm{e}^x
$$

REPLY [3 votes]: HINTS: You have $$\sum_{k=0}^\infty\frac{\cos(k\theta)}{k!}$$
Remember that:
$$e^x=\sum_{k=0}^\infty\frac{x^k}{k!}$$
And:
$$\cos(\theta)=\Re{e^{i\theta}}$$