How do these two summations equate?

Question

Apparently, the summation $$ \sum_{j = i + 1}^n \frac{1}{j - i + 1} $$ is equal to the summation $$ \sum_{k=1}^{n - i} \frac{1}{k + 1} $$ I don't grasp the intuition behind why.

Answer 1

Set $k=j-i$ so when $j=i+1$ then $j-i=1$ so $k=1$. Now when $j=n$ we have $k=j-i=n-i$.

Answer 2

$$ \sum_{j -i = 1}^n \frac{1}{j - i + 1}\\ \text{Let $j-i=k$, giving }\sum_{k=1}^{n-i} \frac{1}{k + 1}$$

Answer 3

$\sum_{j = i + 1}^n \frac{1}{j - i + 1}$

$=\frac{1}{2}+...+\frac{1}{n-i+1}$

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