Talyor series centered at Zo is T(Z) = f(Zo) + f'(Zo)(Z-Zo)^1/1! + f(Zo)(Z-Zo)^2/2! + .... One way to do this is to take the derivative a few times, see if there's a patter, and create the series.

1. cosh z = [e^z + e^-z]/2 so f(z)= (e^2z +1)/2 ; f' = e^2z ; f''(z)= 2e^2z ; f''' = 2^2 e^2z etc. Find each of these at z=0, and we get f(n-prime) = 2^(n-1) . So Taylor series is: Sum[n=0 to infinity]{(1/2)(2z)^(n)/n!}