Prove that $1+\frac12+\frac13+\cdots+\frac1{2^n} \ge 1+\frac{n}{2}$

Question

Is it true that $1+\dfrac12+\dfrac13+ \dots +\dfrac{1}{2^n} \geq 1+\dfrac{n}{2}$?

If so, a proof would be great!

Thank you!

Answer 1

Hint: $$\frac{1}{3}+\frac{1}{4}\ge \frac{1}{4}+\frac{1}{4}=\frac{1}{2}$$

$$\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\ge \frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{1}{2}$$

Answer 2

We have $$1 + \frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\cdots$$ $$\geq 1+\frac{1}{2} + \frac{1}{4} + \frac{1}{4}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\cdots$$ $$=1+\frac{1}{2} + \frac{1}{2} + \frac{1}{2}+\cdots$$ $$=1+\frac{n}{2}.$$