Show that. If $H_{j}=\sum\limits_{k=1}^{j}{\frac{1}{k}}$, than $H_{2^n}\geq 1+\frac{n}{2}$ for all natural numbers.

Question

Show that.

If $H_{j}=\sum\limits_{k=1}^{j}{\frac{1}{k}}$, than

$H_{2^n}\geq 1+\frac{n}{2}$ for all natural numbers.

Answer 1

Hint for every $1\leq k\leq 2^n$ with $n\ge1$ $$\frac{1}{2^n+k}\geq \frac{1}{2^{n+1}} $$

do every term in your sum that is greater than $\frac{1}{2^{n+1}} $ and in the sum there is $2^n$ terms, so the sum is larger than $\frac{1}{2}$.