Fourier expansion of Eisenstein series

Question

In the Fourier expansion of the Eisenstein series (see here), we see that there is the term $$ \sum\limits_{n = 0}^{\infty} q^n.$$ Wikipedia claims that the expansion holds for any $|q| \le 1$. In Serre's book named a course in arithmetic, he defined $q := e^{2\pi iz}$ which has modulus $1$. But for this geometric series to converge, don't we need $|q| < 1$?

Answer 1

You have $$ |q| = \exp(-2\pi \Im(z)),$$ with $\Im(z) > 0$ so $|q| < 1$.