Question

证明，诺limn→∞an=a（a>0）则limn→∞√an=√a？

更一般的limn→∞开k次方an=开k次方a

Answer 1

lim(n-->∞)(1/[n^3+ n]+1/[n^3 +n]+ ... 1/[n^3 +n])
<lim(n-->∞)(1/物巧[n^3+ 1 ]+2/友渣[n^3 +2]+ ... n/[n^3 +n])
<lim(n-->罩告键∞)(n/n^3+n/n^3 + ... n/n^3)
即
0<<lim(n-->∞)(1/[n^3+ 1 ]+2/[n^3 +2]+ ... n/[n^3 +n])<0
故
lim(n-->∞)(1/[n^3+ 1 ]+2/[n^3 +2]+ ... n/[n^3 +n])=0