proof of harmonic series

Question

$s_{2^k}= 1+ \sum_{j=1}^{k}(\sum_{m=2^{k-1}+1}^{2^k} \frac 1m)$ How is this true? For example, when $2^k = 4$, $s_4 = 1+(1/3 +1/4)+(1/3+1/4)$, which is wrong.

Thank you in advance.

Answer 1

This is obviously confusing.$$s_{2^k}= 1+ \sum_{j=1}^{k}(\sum_{m=2^{k-1}+1}^{2^k} \frac 1m)$$

It should have been $$s_{2^k}= 1+ \sum_{j=1}^{k}(\sum_{m=2^{j-1}+1}^{2^j} \frac 1m)$$

The proof is much easier to read without all this sigma stuff.