I want to prove this inequality:
$$ \sum\limits_{i=1}^{2^n} \frac{1}{i} \geq 1 + \frac{n}{2}.$$
So, if I suppose that the inequality holds for a natural number $k$, then
$$ \sum\limits_{i=1}^{2^{k+1}} \frac{1}{i} = {\sum\limits_{i=1}^{2^k} \frac{1}{i}} + {\sum\limits_{i=2^{k}+1}^{2^{k+1}} \frac{1}{i}}.$$
Thus, I just have to prove that $\sum\limits_{i=2^{k}+1}^{2^{k+1}} \frac{1}{i} \geq \frac{1}{2}$, but I'm stuck on this. I know there are $2^{k}$ natural numbers bewteen $2^k$ and $2^{k+1}$, but I'm not sure how to use it . I'd appreciate your help.
Answer
It's enough to prove that
$$1+\frac{n}{2}+\frac{1}{2^n+1}+\frac{1}{2^n+2}+...+\frac{1}{2^{n+1}}\geq1+\frac{n+1}{2}$$ or
$$\frac{1}{2^n+1}+\frac{1}{2^n+2}+...+\frac{1}{2^{n+1}}\geq\frac{1}{2},$$
which is true because
$$\frac{1}{2^n+1}+\frac{1}{2^n+2}+...+\frac{1}{2^{n+1}}\geq\frac{1}{2^{n+1}}+\frac{1}{2^{n+1}}+...+\frac{1}{2^{n+1}}=\frac{2^n}{2^{n+1}}=\frac{1}{2}.$$
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