Question

1.求limx(n),其中x(1)>0,x(n+1)=[2+3x(n)]/[1+x(n)],n=1,2,……（）代表下标

Answer 1

1 x(n+1)=[2+3x(n)]/[1+x(n)]=3 - 1/(1+x(n) )

x(1)>0 3>x(2)>0 可归纳的 3>x(n)>0

易知 x(n) 单调递增有界 设limx(n)=a

x(n+1)=3 - 1/ (1+x(n) ) 两边取极限

a=3-1/(1+a)解得a=1+√3

2 拆分2/(n+1)(n+2)(n+3)]=1/(n+1)-2/(n+2)+1/(n+3)

2/(2*3*4)=1/2-2/3+1/4 拆后有前后消掉有

2/(2*3*4)+2/(3*4*5)+……2/(n+1)(n+2)(n+3)]=1/2-1/3-1/(n+2)+1/(n+3)

lim[2/(2*3*4)+2/(3*4*5)+……2/(n+1)(n+2)(n+3)]=1/6