Does the demonstration of Euler's formula via power series require said power series to be proven to converge over complex domain

Question

Stackexchance users. I would apprechiate some help with following beginner issue:

So, the demonstration of Euler's formula by comparing Maclaurin series of e^(iz), cos(z) and sin(z).

Assuming I have the real-valued Maclaurin series's at hand, are they sufficient to be used on this demonstration? Or do I need to prove the convergence over complex domain first (which would make the whole process insufficient for my purpose, which is to show that e^z is holomorphic over entire complex domain).