Question

if Sn is the nth partial sum of the harmonic series, show that e^Sn > n+1. Why does this imply that the harmonic series is divergent?

Answer 1

`S_n=\displaystyle\sum_(k=1)^n\frac1k`

You have


`\ln e^(S_n)=S_n\ln e=S_n`

so showing that
`e^(S_n)>n+1` amounts to the same as showing that
`S_n>\ln(n+1)`.

As
`n\to\infty`, you have
`\ln(n+1)\to\infty`. By comparison, then, it follows that
`S_n\to\infty` at a faster rate, which means
`S_\infty` must diverge.