Just one line is giving me trouble:
$(2^{-l+1} + 2^{-l} + 2^{-l-1}+ \cdots + 2^{-m+2} )\cdot M$
$=[(2^{-l+2} - 2^{-l+1}) + (2^{-l-1} - 2^{-l}) + \cdots +(2^{-m+3} - 2^{-m+2}) ]\cdot M$
Hows that?
Just one line is giving me trouble:
$(2^{-l+1} + 2^{-l} + 2^{-l-1}+ \cdots + 2^{-m+2} )\cdot M$
$=[(2^{-l+2} - 2^{-l+1}) + (2^{-l-1} - 2^{-l}) + \cdots +(2^{-m+3} - 2^{-m+2}) ]\cdot M$
Hows that?
Its from the fact that $2^{k+1} - 2^k = 2^k(2-1) = 2^k$ Therefore each power of $2$ is expanded as a difference.
i.e. $2^{-l+1} = 2^{-l+2} - 2^{-l+1}$ and so on....
Oh, your comment made me see! Check that for any $\,n\in\Bbb N\,$:
$2^n-2^{n-1}=2^{n-1}\left(2-1\right)=2^{n-1}...!!$
so $\,2^{-l+1}=2^{-l+2}-2^{-l+1}\,$ and etc.