Question

如图，这两个求数列极限，是用夹逼吗？该怎么解？

Answer 1

(2)

n/(n+lnn)≤1/(n+ln1)+1/(n+ln2)+...+1/(n+lnn)≤n/(n+ln1)
lim(n->无穷) n/(n+lnn) = lim(n->无穷) n/(n+ln1) =1
=>
lim(n->无穷) [ 1/(n+ln1)+1/(n+ln2)+...+1/(n+lnn) ] =1
(3)
e^1+e^2+...+e^n
= e( e^n -1)/(e-1)
=>
e( e^n -1)/[(e^n +n^2)(e-1)]≤e/(e^n+1^2)+e^2/(e^n+2^2)+...+e^n/(e^n+n^2)
and
e/(e^n+1^2)+e^2/(e^n+2^2)+...+e^n/(e^n+n^2)≤ e( e^n -1)/[(e^n +1^2)(e-1)]
lim(n->无穷) e( e^n -1)/[(e^n +n^2)(e-1)] = e/(e-1)
lim(n->无穷) e( e^n -1)/[(e^n +1^2)(e-1)] = e/(e-1)
=>
lim(n->无穷) [ e/(e^n+1^2)+e^2/(e^n+2^2)+...+e^n/(e^n+n^2) ]
= e/(e-1)

Answer 2

对的，要用夹逼性
就是把分母都放到最大算一下，再把分母都放到最小算一下