Can someone make this question clear to me and give me a hint?

Question

By using induction, prove that $s_{2^n} \geq 1 +n/2$ for all $n\in \mathbb N$, where $s_j=\sum\limits_{i=1}^j 1/i$ is the $j$-th partial sum of the Harmonic Series. Note that this implies the partial sums diverge, which proves that the Harmonic Series diverges.

Answer 1

Hint: for $i=2^n, 2^n+1, \ldots, 2^{n+1}-1$, you have:

$$\frac{1}{i}>\frac{1}{2^{n+1}},\quad 2^n\leq i<2^{n+1}$$

This will prove the fact that $\sum_1^\infty 1/n = \infty$, because $\sum_1^\infty 1/n = \lim_{n\to\infty} s_n$, and we will have:

$$s_1 = \frac11 = 1$$ $$s_2 = \frac11 + \frac12 = 1 + \frac12$$ $$s_4 = \frac11 + \cdots + \frac14 \geq 1 + 2/2$$ $$s_8 = \frac11 + \cdots + \frac18 \geq 1 + 3/2$$ $$s_{16} = \frac11 + \cdots + \frac1{16} \geq 1 + 4/2$$

So that the sequence $s_n$ will grow arbitrarily large (although slowly)