lim(1+1/2+1/4+…+1/2^n)/(1+1/3+1+9+…+1/3^n)x→∞过程尽量详细点!

Question

l im (1+1/2+1/4+…+1/2^n) / (1+1/3+1+9+…+1/3^n) x→∞
过程尽量详细点!

Answer 1

因为l im_{ n→∞} 1+1/2+1/4+…+1/2^n = 1/(1-1/2) =2,
l im_{ n→∞} 1+1/3+1/9+…+1/3^n = 1/(1-1/3) =3/2 ≠0,
故 l im_{ n→∞} (1+1/2+1/4+…+1/2^n) / (1+1/3+1/9+…+1/3^n)
= ( l im_{ n→∞} 1+1/2+1/4+…+1/2^n ) / ( l im_{ n→∞} 1+1/3+1/9+…+1/3^n )
=2 / (3/2)
= 4 / 3