From: xxyzz <xxyzz@idt.net>
Subject: Re: Need help with an infinite series
Date: Sat, 17 Jul 1999 23:47:55 -0400
Newsgroups: sci.math
Keywords: Mahler functions

"G. A. Edgar" wrote:
> 
> In article <7mqtdj$i00$1@nntp3.atl.mindspring.net>, Bob Harris
> <nitlion@mindspring.com> wrote:
> 
> > Oops, see corrections below.
> >
> > > Howdy,
> > >
> > > Can anyone here give me some clues on how to solve this infinite series:
> > >
> > > F(x) := sum(x^(2^k),k=0..infinity)
> > >
> >   That is, F(x) = x + x^2 + x^4 + x^8 + x^16 + x^32 + ....     <<<
> 
> This is not an elementary function.  There has been some
> study of its nature as a function of a complex variable... it
> converges for all complex |x| < 1.  If I remember correctly,
> the zeros accumulate to the entire unit circle.
> 

This is an example of what is called (well at least by people in
transcendental number theory) a Mahler Function.  If you want to study
such functions, you might want to look at Kumiko Nishioka's book Mahler
Functions and Transcendence, Lecture Notes in Math #1631, published by
Springer-Verlag.