How to show the following inequality, $H_{2^n}\geq n/2$?

Question

Let $H_n=\sum_{k=1}^{n} 1/k $

I want to show that $H_{2^n}\geq n/2$ for every $n$ greater than or equal to 1.

How should I proceed?

Answer 1

For instance by induction.

Hint for the inductive step:

$$H_{2^{n+1}}=H_{2^{n}}+\frac1{2^n+1}+\dots+\frac1{2^{n+1}}=H_{2^{n}}+\underbrace{\frac1{2^n+1}+\dots+\frac1{2^n+2^n}}_{2^n\:\text{terms}}.$$