Question

an,bn都是公差不为零的等差数列,lim(an/bn)=3,求limb1+b2+.+bn/na3n

Answer 1

设an,bn公差分别为d1,d2,因为lim(an/bn)=lim(a1+(n-1)d1)/(b1+(n-1)d2)=3,所以d1/d2=3

b1+b2+.+bn=(b1+bn)n/2=(2b1+(n-1)d2)n/2

a3n=a1+(3n-1)d1

所以lim(b1+b2+.+bn/na3n)=lim(2b1+(n-1)d2)/(2*(a1+(3n-1)d1))

=lim(n-1)d2/2(3n-1)d1=lim(n-1)/6(3n-1)=1/18