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\begin{align*} &\sum_{k=1}^{n} \frac{2^k-1}{k} \\ ={}&\sum_{k=1}^{n} \frac{1}{k}\left(\sum_{j=1}^{k} \binom{k}{j}\right) \\ ={}&\sum_{j=1}^{n} \sum_{k=j}^{n} \binom{k}{j}\frac{1}{k} \\ ={}&\sum_{j=1}^{n} \frac{1}{j}\left(\sum_{k=j}^{n} \binom{k-1}{j-1}\right) \\ ={}&\sum_{j=1}^{n} \frac{1}{j} \binom{n}{j}. \end{align*}\begin{align*} \sum_{k=1}^{n} \frac{2^k-1}{k} &=\sum_{k=1}^{n} \frac{1}{k}\left(\sum_{j=1}^{k} \binom{k}{j}\right) \\ &=\sum_{j=1}^{n} \sum_{k=j}^{n} \binom{k}{j}\frac{1}{k} \\ &=\sum_{j=1}^{n} \frac{1}{j}\left(\sum_{k=j}^{n} \binom{k-1}{j-1}\right) \\ &=\sum_{j=1}^{n} \frac{1}{j} \binom{n}{j}. \end{align*}

$\sum_{k=1}^{n} \frac{2^k-1}{k}$

$=\sum_{k=1}^{n} \frac{1}{k}(\sum_{j=1}^{k} \binom{k}{j})$

$=\sum_{j=1}^{n} \sum_{k=j}^{n} \binom{k}{j}\frac{1}{k}$

$=\sum_{j=1}^{n} \frac{1}{j}(\sum_{k=j}^{n} \binom{k-1}{j-1})$

$=\sum_{j=1}^{n} \frac{1}{j} \binom{n}{j}$\begin{align*} &\sum_{k=1}^{n} \frac{2^k-1}{k} \\ ={}&\sum_{k=1}^{n} \frac{1}{k}\left(\sum_{j=1}^{k} \binom{k}{j}\right) \\ ={}&\sum_{j=1}^{n} \sum_{k=j}^{n} \binom{k}{j}\frac{1}{k} \\ ={}&\sum_{j=1}^{n} \frac{1}{j}\left(\sum_{k=j}^{n} \binom{k-1}{j-1}\right) \\ ={}&\sum_{j=1}^{n} \frac{1}{j} \binom{n}{j}. \end{align*}

$$\sum_{k=1}^{n} \frac{2^k-1}{k}$$$\sum_{k=1}^{n} \frac{2^k-1}{k}$

$$=\sum_{k=1}^{n} \frac{1}{k}(\sum_{j=1}^{k} \binom{k}{j})$$$=\sum_{k=1}^{n} \frac{1}{k}(\sum_{j=1}^{k} \binom{k}{j})$

$$=\sum_{j=1}^{n} \sum_{k=j}^{n} \binom{k}{j}\frac{1}{k}$$$=\sum_{j=1}^{n} \sum_{k=j}^{n} \binom{k}{j}\frac{1}{k}$

$$=\sum_{j=1}^{n} \frac{1}{j}(\sum_{k=j}^{n} \binom{k-1}{j-1})$$$=\sum_{j=1}^{n} \frac{1}{j}(\sum_{k=j}^{n} \binom{k-1}{j-1})$

$$=\sum_{j=1}^{n} \frac{1}{j} \binom{n}{j}$$$=\sum_{j=1}^{n} \frac{1}{j} \binom{n}{j}$