Sum of Scaled Harmonic Numbers

Using Iverson Brackets can simplify the changing of the order of summation: $$ \begin{align} \frac1n\sum_{j=1}^n\sum_{k=j}^n\frac1k &=\frac1n\sum_{j=1}^n\sum_{k=1}^n[k\ge j]\frac1k\\ &=\frac1n\sum_{k=1}^n\sum_{j=1}^n[k\ge j]\frac1k\\ &=\frac1n\sum_{k=1}^n\sum_{j=1}^k\frac1k\\ &=\frac1n\sum_{k=1}^n1\\[9pt] &=1 \end{align} $$

In the double sum $$\sum_{j=1}^{n}\sum_{k=j}^{n}\frac{1}{k}$$ each fraction $1/k$ occurs $k$ times, so overall these contribute $1$ to the sum. As $k$ ranges from $1$ to $n$, the total sum is $n$.

We can also write the index region conveniently to better see what's going on.

We obtain \begin{align*} \frac{1}{n}\sum_{j=1}^n\sum_{k=j}^n\frac{1}{k}&=\frac{1}{n}\sum_{\color{blue}{1\leq j\leq k\leq n}}\frac{1}{k}\\ &=\frac{1}{n}\sum_{k=1}^n\sum_{j=1}^k\frac{1}{k}\\ &=\frac{1}{n}\sum_{k=1}^n1\\ &=1 \end{align*}