Sn+1=Sn+a(n+1)=2Sn+a1
Sn=a(n+1)-a1
Sn-1=an-a1
an=Sn-Sn-1=a(n+1)-an
a(n+1)=2an
a(n+1)/an=2,为定值.
数列{an}是以a1为首项,2为公比的等比数列.
an=a1×2^(n-1) 2an=a1×2ⁿ
2an/Sn=a1×2ⁿ/[a1×(2ⁿ-1)/(2-1)]=2ⁿ/(2ⁿ-1)=1/(1- 1/2ⁿ)
当n->+∞时,1/2ⁿ->0 1-1/2ⁿ->1 2an/Sn->1
lim(2an/Sn)=1