数列的极限3

来源：百度知道 编辑：UC知道 时间：2024/07/04 02:15:07
一正数数列是等比数列，a2=4,a4=16
求lim (lg an+1 + lg an+2 + .....+ lg a2n)/n②

过程详细，做法越简单越好..

A[n]=2^n;
lg( A[n+1])=log2(A[n+1])/log2(10)=(n+1)/log2(10)
所以lim (lg an+1 + lg an+2 + .....+ lg a2n)/n^2=
lim[(n+1)+(n+2)+ ...+(n+n)/(n^2*log2(10))]=
lim[(1+1/2*(1+1/n))/log2(10)]=3/2/log2(10)=3/2*lg2

a2＝4，a4＝16，则公比r^2＝16/4＝4，r＝2，a1＝a2/r＝2，所以an＝2×2^(n-1)＝2^n

[lg(a(n+1))＋lg(a(n+2))＋……＋ln(a(2n))]/n^2
＝2lg2×[(n+1)＋(n+2)＋……＋(2n)]/n^2
＝2lg2×n(3n+1)/2n^2
＝2lg2×(3＋1/n)/2→2lg2×3/2＝3lg2