I am trying exercises in complex analysis from tutorials of an Institute of which I am not a student.

There is a question in infinite products on which I am struct.

Question is - Prove that infinite product $$\prod_{n=1}^{\infty} (1- e^{2πin\tau} )$$ , $$\tau$$ belongs to Upper Half plane converges absolutely.

I have studied complex analysis from Ponnusamy and silvermann and from that I know that $$\prod_{n=1}^{\infty} (1+ a_n )$$ converges absolutely iff $$\sum_{n=1}^{\infty} (a_n)$$ converges absolutely. But I don't know how to prove absolute convergence of sum $$e^{2πin\tau }$$ .

Can someone please explain how to do this

Hint: Write $$\tau = a +ib.$$ Then $$b>0.$$ We have

$$e^{2\pi i n\tau} = e^{-2\pi nb}\cdot e^{2\pi i na}.$$

What is the absolute value of the last expression?

• It's always less than <1 hence absolutely convergent. – Ben Jan 11 '20 at 20:46