Find the sum of the series  $$ \sum_{n=0}^{\infty} \frac{1}{n ! 2^{n}} \cos \frac{\pi n-1}{2} . $$
For all $x \in \mathbb{R}$, it holds  $$ \begin{aligned} &\sum_{n=0}^{\infty} \frac{\cos \left(\frac{\pi n}{2}-x\right)}{n !} x^{n}=\cos x \sum_{n=0}^{\infty} \frac{\cos \frac{\pi n}{2}}{n !} x^{n}+\sin x \sum_{n=0}^{\infty} \frac{\sin \frac{\pi n}{2}}{n !} x^{n}= \\ &=\cos x \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2 k) !} x^{2 k}+\sin x \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2 k+1) !} x^{2 k+1}=\cos ^{2} x+\sin ^{2} x=1 . \end{aligned} $$  In particular for $x=\frac{1}{2}$ we obtain the sum of the series in question.  Answer: 1