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_bare 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.