How do we justify splitting a sum over $1 \le j < k+j \le n$ into two sums $1 \le k \le n$ and $1 \le j \le n-k$?

Question

Can anyone justify the following summation manipulation?

$$\sum_{1 \le j < k+j \le n} \frac{1}{k} = \sum_{1 \le k \le n}\,\sum_{1 \le j \le n-k} \frac{1}{k}$$

Answer 1

Firstly, we check the range of $j$ for each $k$:

RHS says for each $k$ we have $1\leq j\leq n-k$.

LHS: Since $k+j\leq n \iff j\leq n-k$, the LHS also has $1\leq j\leq n-k$.

$$\\$$

Now we check the range of $k$:

LHS: $j\lt k+j \implies 1\leq k$. Also, $1\leq j$ and $k+j\leq n \implies k\leq n-1$. So the range of $k$ is $1\leq k \leq n-1$.

RHS has $1 \leq k \leq n$. But this is effectively $1 \leq k \leq n-1$ since with $k=n$ in the outer sum, the inner sum has no terms and equals $0$.