I am self studying Analytic number theory from Tom M Apostol and I cannot think about a deduction in proof of a theorem in Chapter 6 ( Modular forms with multiplicative coefficients) .
In section 6.15 - Estimates for the Fourier coefficients of Entire forms
Apostol assumes f to be an entire form with Fourier expansion
f($\tau$) = $\sum_{n=0}^\infty = c(n) x^n $ . ----(1)
Then later in same page Apostol mentions
The series in (1) converges absolutely if |x| <1 .
I am unable to think why f($\tau$) converges absolutely for |x|<1 although there is no restriction on coefficients c(n) ? Why does value of coefficients c(n) doesn't matters and what theorem or result is Apostol actually using to deduce this.
My second doubt is -- Apostol later writes If $\tau $ is in fundamental region of $\Gamma$ then $\tau$ = u +iv with v $\ge$ $\frac {√3} {2} $
Can somebody please tell why v couldn't be less than $\frac {√3} {2} $ ?