<s>[INST] How to show that $ \sum_{k=1}^{2^n}\frac{1}{k}\geq 1+\frac{n}{2}$? So in order to prove that the Harmonic series diverge, I want to show that:
$$ \sum\limits_{k=1}^{2^n}\dfrac{1}{k}\geq 1+\dfrac{n}{2}$$
It is clear that if we expand the sum that this inequality holds true,
$$n=3,\ \sum\limits_{k=1}^{2^3=8}\dfrac{1}{k}=1+\frac{1}{2}+\underbrace{\frac{1}{3}+\frac{1}{4}}\limits_{\geq \frac{1}{2}}+\underbrace{\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}}\limits_{\geq \frac{1}{2}}$$
but how can i show that using induction?
Any tips would be helpful, and thanks.
 [/INST] *

*Use the fact that $\frac{1}{3}+\frac{1}{4} \geq \frac{1}{4}+\frac{1}{4}$ and $\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8} \geq \frac{1}{8}+\frac{1}{8} +\frac{1}{8}+\frac{1}{8}$

*There are also various other proofs which you might be interested to see. Here is a link: http://scipp.ucsc.edu/~haber/archives/physics116A10/harmapa.pdf
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