Question

已知等比数列an中a1=1,公比为x(x〉0)其前n项和为Sn,bn=an/Sn,求lim(n→○○)bn

Answer 1

an为等比数列

an=a1*q^(n-1)=x^(n-1)

sn=a1(q^n-1)/(q-1)=(x^n-1)/(x-1)

那么,

bn=an/sn=(x-1)x^(n-1)/(x^n-1)

那么,

n→∞

lim bn

=lim (x-1)x^(n-1)/(x^n-1)

=lim (x-1)x^(n-1) / (x-1)(x^(n-1)+x^(n-2)+…+x^2+x+1)

=lim x^(n-1)/(x^(n-1)+x^(n-2)+…+x^2+x+1)

上下同时除以x^(n-1)

=lim 1/(1+1/x+…+1/x^(n-1))

=lim 1/ [1*((1/x)^n-1)/((1/x)-1)]

=lim ((1/x)-1) / ((1/x)^n-1)

=((1/x)-1) / (-1)

=1-1/x

=(x-1)/x

有不懂欢迎追问