Bypassing dynamical systems : A simple way to get the box-counting dimension of the graph of the Weierstrass function

In the following, bypassing dynamical systems tools, we propose a simple means of computing the box dimension of the graph of the classical Weierstrass function defined, for any real number~$x$, by~$ {cal W}(x)=displaystyle sum_{n=0}^{+infty} lambda^n,cos left ( 2, pi,N_b^n,x right) $, where~$lambda$ and~$N_b$ are two real numbers such that~mbox{$0 <lambda<1$},~mbox{$ N_b,in,N$} and~$ lambda,N_b > 1 $, using a sequence a graphs that approximate the studied one.