求极限lim(1/2n+3/4n+……+(2^n-1)/(2^n*n))

1/(2n) + 3/(4n) +…… + (2^n-1)/(2^n*n))
= [(1-1/2) + (1- 1/4) + (1- 1/8) + .+ (1- 1/2^n) ] / n
= 1 - ( 1/2 + 1/4 + 1/8 + 1/2^n ) / n
= 1 - [ 1/2 - 1/2^(n+1)] / (1-1/2) / n
0 < [1/2 - 1/2^(n+1)] / (1-1/2) / n < [1/2 / (1-1/2) ] /n = 1/n -> 0
原式 = 1 - 0 = 1