# How do I find the sum of $\frac{\cos(n)}{n!}$?

I am having a hard time to calculate the sum of $$\sum_{n\geq 0} \frac{\cos(n)}{n!}$$ I don't know how to approach this. Could you give any thints for this?

• $\cos n = \operatorname{Re} ( \cos n + i\sin n)$ – Daniel Fischer Oct 25 '16 at 15:50
• I'm surprised you're dealing with $\cos n$ for integer $n$ - usually arguments to cosine are either continuous or integer multiples of $\pi$ - it's unusual, say, to run into $\sin 1$. Is this really the problem you meant to ask? – QuantumFool Oct 25 '16 at 15:51
• @QuantumFool, expressions such as $\sin n$ and $\cos n$ for $n \in \Bbb{Z}$ aren't that uncommon when dealing with infinite series in an academic setting. – user307169 Oct 25 '16 at 15:57
• @QuantumFool, expressions such as $\sin n$ and $\cos n$ for $n \in \Bbb{Z}$ aren't that uncommon when dealing with infinite series in an academic setting. – user307169 Oct 25 '16 at 15:57

## 1 Answer

\begin{align} \sum\frac{\cos n}{n!}& =\Re\sum\frac{e^{in}}{n!} \\ & = \Re e^{e^i}\\ & = e^{\cos 1} \cos(\sin 1) \end{align}

• nice solution! +1) – haqnatural Oct 25 '16 at 15:59
• Short and clear! Perfect, thank you! – John Mayne Oct 25 '16 at 17:19