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.