(1)当n=1时,显然不等式成立.
(2)假设当n=k∈N*时,不等式成立即1+1/根号2+1/根号3+……+1/根号<2根号n,则当n=k+1时有:1+1/根号2+……+1/根号n+1/根号(n+1)<2根号n+1/根号(n+1)
将2根号n+1/根号(n+1)通分然后乘以根号(n+1)然后减1并平方得:4n^2+4n
将2根号(n+1)乘以根号(n+1)然后减1并平方得4n^2+4n+1
则2根号n+1/根号(n+1)<2根号(n+1)
即1+1/根号2+……+1/根号n+1/根号(n+1)<2根号n+1/根号(n+1)<2根号(n+1)则不等式也成立.
由(1)(2)得对任意n∈N*都有1+1/根号2+1/根号3+……+1/根号n