I'm trying to show that the Harmonic series diverges, using induction. So far I have shown: If we let $s_{n} = \sum_{k=1}^{n}\frac{1}{k}$
- $s_{2n} \geq s_{n} + \frac{1}{2}, \forall n$
- $s_{2^{n}} \geq 1 + \frac{n}{2}, \forall n$ by induction
The next step is to deduce the divergence of $\sum_{n=1}^{\infty}\frac{1}{n}$. I know that it does diverge but I don't directly see how the above two parts help.
Can I just say that since $s_{2^{n}} \geq 1 + \frac{n}{2}$ then as n tends to infinity, the partial sums of $s_{2_{n}}$ diverge? But that's for $s_{2^{n}}$ and not $s_{n}$?
Since $(\forall n\in\mathbb N):s_{2^n}\geqslant1+\frac n2$ and since $\lim_{n\to\infty}1+\frac n2=\infty$, $\lim_{n\to\infty}s_{2^n}=\infty$. Therefore, the sequence $(s_n)_{n\in\mathbb N}$ diverges, since its subsequence $(s_{2^n})_{n\in\mathbb N}$ diverges.