# Find sum of $1 + \cos \theta + \frac{1}{2!}\cos 2\theta + \cdots$

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.

• Hint: $\cos(n\theta) = \Re( e^{i n\theta})$ for $\theta \in \mathbb{R}$. Aug 14, 2015 at 15:00

Hint: $$1+\cos x + \frac{1}{2!}\cos 2x + \ldots = \Re(e^{0ix} + e^{1ix} + \frac{1}{2!}e^{2ix} + \ldots)=\Re e^{e^{ix}}$$

$$e^{ix}=\cos x+i\sin x\\\Longrightarrow e^{e^{ix}} = e^{\cos x}e^{i\sin x}=e^{\cos x}(\cos(\sin x)+i\sin(\sin x))$$ Your sum is $$e^{\cos x}\cos(\sin x)$$

• Spoilers with \$\$ doesn't work properly Aug 14, 2015 at 15:19

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: $$\cos(\theta)=\Re{e^{i\theta}}$$

• Is $\cos\theta$ really equal to $e^{i\theta}$? Aug 14, 2015 at 15:04
• Then what is $\cos\theta+i\sin\theta$? Aug 14, 2015 at 15:05
• @AdityaAgarwal typo fixed Aug 14, 2015 at 15:05

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 $$\lim_{n\to\infty}\sum_{k=0}^n\frac{x^k}{k!}=\mathrm{e}^x$$