1/2+2/4+3/8+4/16+5/32+6/64+7/128+8/256+9/512+10/1024 +11/204

令s=1/2+2/4+3/8+.+n/2^n
s=1/2+2/2^2+3/2^3+.+n/2^n
s/2=1/2^2+2/2^3+3/2^4+.(n-1)/2^n+n/2^(n+1)
s-s/2=1/2+1/2^2+1/2^3+1/2^4+.+1/2^n-n/2^(n+1)
s/2=1/2*[1-(1/2)^n]/(1-1/2)-n/2^(n+1)
s/2=1-1/2^n-n/2^(n+1)
s/2=1-2/2^(n+1)-n/2^(n+1)
s/2=1-(n+2)/2^(n+1)
s=2-(n+2)/2^n
即1/2+2/4+3/8+4/16+5/32+6/64+7/128+8/256+9/512+10/1024 +11/2048+.+n/2^n=2-(n+2)/2^n