How many terms should be used in taylor series expansion of the function f(z) = e^z around z = 0 for a specific value of z = 30 + 30 i to get an error of less than 0.05 using Mathematica?
Poly[x_ , n_] := Series[Exp[x] , {x,0,n}]
errorTaylor[x_ , n_] := (1/Factorial[n])*Integrate[((x-t)^n)*Exp[t],{t,0,x}]
Reduce[{errorTaylor[30+30 I,n]<0.05},n,Complexes]
The output was
°°° Reduce Reduce was unable to solve the system with inexact coefficients or the system obtained by rationalization of inexact numbers present in the system. Since many of the methods used by Reduce require exact input, providing Reduce with an exact version of the system may help.
Reduce[{e^(30+30 I)(Gamma[1+n]-Gamma[1+n,30+30 I])/(n!)<0.05 if Re[n]>-1},n, **C**]