$\cosh z$ keeps exactly the even terms of the exponential series.Suggest a slogan
For every \(z\in\CC\), \[ \cosh z=1+\frac{z^{2}}{2 !}+\frac{z^{4}}{4 !}+\frac{z^{6}}{6 !}+\cdots=\sum_{n=0}^{\infty} \frac{z^{2 n}}{(2 n) !}. \]
What is the series expansion of $\cosh(z)$?