$\sinh z$ keeps exactly the odd terms of the exponential series.Suggest a slogan
For every \(z\in\CC\), \[ \sinh z=z+\frac{z^{3}}{3 !}+\frac{z^{5}}{5 !}+\frac{z^{7}}{7 !}+\cdots=\sum_{n=0}^{\infty} \frac{z^{2 n+1}}{(2 n+1) !}. \]
What is the series expansion of $\sinh(z)$?