求证16小于号(Σ（上标k=1,下标80）1/根号k）大于号17即16小于号（1/根号1+1/根号2+1/根号3..+1/根号80）大于号17

Question

求证16小于号(Σ（上标k=1,下标80）1/根号k）大于号17
即16小于号（1/根号1+1/根号2+1/根号3..+1/根号80）大于号17

Answer 1

1+1/√2+1/√3+...+1/√80
=1+2/2√2+2/2√3+...+2/2√80
>1+2[1/(√2+√3)+1/(√3+√4)+...+1/(√80+√81)]
=1+2(√3-√2+√4-√3+...+√81-√80)
=1+2√81-2√2
=19-2√2
=16.17>16
1/√K