A Function An Integral

$\large \lim_{k \to \infty} \int_{-\infty}^\infty \left(\frac {|x-1|}2 + \frac {|x+1|}2 - |x| + \sum_{n=2}^{2k+1} (-1)^n \sum_{j=1}^{2^{n-1}} \frac {\left|-\frac {2^{n-1}-1}{2^{n-1}}+\frac {j-1}{2^{n-2}}+x \right|}{2^n} \right) dx = \frac AB$

where $A$ and $B$ are relatively prime positive integers. Submit $A+B$ .