数列问题6

an=n/(n+2)+(n+2)/n
=[n^2+(n+2)^2]/[n(n+2)]
=[2n(n+2)+4]/[n(n+2)]
=2+4/[n(n+2)]
=2+2[1/n-1/(n+2)]

Sn=a1+a2+……+an
=2n+2[1-1/3+1/2-1/4+1/3-1/5-……+1/(n-1)-1/(n+1)+1/n-1/(n+2)]
=2n+2[1+1/2-1/(n+1)-1/(n+2)]
=2n+3-2[1/(n+1)+1/(n+2)]
2/(n+1)+2/(n+2)≤2(1/2+1/3)=5/3
Sn≥2n+3-5/3>2n
1/(n+1)+1/(n+2)>0
Sn<2n+3

2n<Sn<2n+3

整理An=2+2[1/n-1/(n+2)]

Sn＝2n＋2[(1-1/3)+(1/2-1/4)+(1/3-1/5)+...+1/(n-1)-1/(n+1)+1/n-1/(n+2)]
＝2n＋2[1＋1/2－1/(n+1）－1/(n+2)]＝2n+3－2[1/(n+1）+1/(n+2)]

而0<[1/(n+1）+1/(n+2)]<<3/2 所以 2n<Sn<2n+3