real analysis – one summation and related asymptotic of not well known series

I have a summation for one of my research as follows:

sum of gathered bucket $B= frac{1}{2} * 1 + frac{1}{4} * 2 + frac{1}{8} * 3 + … + frac{1}{2^{n-2}} * (n-2) + frac{1}{2^{n-1}} * (n-1) + frac{1}{2^{n-1}} * (n-1) $

I need an asymptotic notation for $B$. How I can summarize this sum to
find $O$ equivalency class or big-o notation.