Coefficient of $x^{2^{m+1}}$ in the expansion of $E=\frac{1}{(1+x)(1+x^2)(1+x^4)(1+x^8)\cdots (1+x^{2^{m,}})}(\left|x\right|<1)$  is 

$E={(1+x)}^{-1}{(1+x^2)}^{-1}{(1+x^4)}^{-1}\cdots {(1+{x^2}^{{ }^m})}^{-1}$

$=(1-x+x^2\cdots )(1-x^2+\cdots )(1-x^4\cdots )\cdots (1-{x^2}^{{ }^m}+\cdots )$

Here highest power of x is $x.x^2\cdots x^{2^m}$

$\begin{array}{l}=x^{1+2+2^2+\cdots 2^m}\\ =x^{\left(\frac{2^{m+1}-1}{2-1}\right)}\\ =x^{2^{m+1}-1}<x^{2^{m+1}}\end{array}$

$\therefore x^{2^{m+1}}$ term does not exist

$\therefore$ coefficient $=0$