Infinite Sum Practice

We will use the Euler's formula ( $e^{i\theta}=\cos \theta+i\sin \theta$ ) a good many times while arriving at the desired closed form.

We will also use the Maclaurin series for $e^x \ \ \rightarrow \ \ \displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{n!}=e^x$ .

Let us begin our solution $\rightarrow$

$\begin{aligned} S &= \dfrac{\cos 0}{e^0 \cdot 0!}+ \dfrac{\cos 1}{e^1 \cdot 1!}+\dfrac{\cos 2}{e^2 \cdot 2!}+ \cdots \cdots \\ &= \displaystyle\sum_{n=0}^{\infty} \dfrac{\cos n}{e^n \cdot n!}\\ &= \Re \left\{ \displaystyle\sum_{n=0}^{\infty} \dfrac{e^{in}}{e^n \cdot n!}\right\}\\ &= \Re \left\{ \displaystyle\sum_{n=0}^{\infty} \dfrac{e^{in-n}}{n!}\right\}\\ &=\Re \left\{ \displaystyle\sum_{n=0}^{\infty} \dfrac{\left(e^{i-1}\right)^n}{n!}\right\}\\ &=\Re \left\{e^{e^{i-1}}\right\}\\ &=\Re \left\{e^{e^i/e}\right\}\\ &=\Re \left\{e^{(\cos 1+i\sin 1)/e}\right\}\\ &=\Re \left\{e^{(\cos 1)/e} \cdot e^{i((\sin 1)/e)}\right\}\\ &= \Re \left\{{e^{(\cos 1)/e}}\left(\cos \left(\dfrac{\sin 1}{e}\right)+i\sin \left(\dfrac{\sin 1}{e}\right)\right)\right\}\\ &= e^{(\cos 1)/e} \cos \left(\dfrac{\sin 1}{e}\right)\\ &\approx \boxed{1.16} \end{aligned}$