lim(1\2+3\2^2+5\2^3...+2n-1\2^n)(n→∞)的极值

先求和
Sn = 12+32^2+52^3+...+(2n-3)2^(n-1)+(2n-1)2^n
2Sn= 1+32+52^2+72^3+...+(2n-1)2^(n-1)
下减上
Sn=1+22+22^2+22^3+...+22^(n-1)-(2n-1)2^n
=1+(1+1/2+...+1/2^(n-1))-(2n-1)/2^n
=1+1(1-(1/2)^n)/(1-1/2)-(2n-1)/2^n
=3-(1/2)^(n-1)-(2n-1)/2^n
当n->无穷
Sn->3+0+0=3
limn->无穷(1/2)^(n-1)=0 显然
limn->无穷(2n-1)/2^n=limx->无穷(2x-1)/2^x
洛必达=lim x->无穷 2/2^xln2=0
所以
lim(12+32^2+52^3...+2n-12^n)(n→∞)
=3

3