limn-->∞(1+1/1+2+1/1+2+3+...+1/1+2+3+...+n)求极限及过程,

注意一般项 为1/(1+2+3+....+k) = 2/(k+1)k = 2(1/k - 1/k+1)
所以上式可化为2(1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 +....+ 1/n - 1/(n+1))
=2(1-1/(n+1)) 当n趋于无穷大时 极限为2
2

1/1+2+3+...n=2/n(n+1)
=lim(2/1*2+2/2*3+2/3*4+....2/n(n+1)]=2lim(1-1/2+1/2-1/3+1/3-1/4+...1/n-1/(n+1)]
=2

1/(1+2+……+n)=1/[n(n+1)/2]=2/[n(n+1)]=2*[1/n-1/(n+1)]
所以极限内的式子=2*{(1/1-1/2)+(1/2-1/3)+……+[1/n-1/(n+1)]]
=2*(1-1/(n+1)]
n→∞,1/(n+1)→0
所以极限=2*(1-0)=2