How do these two summations equate?

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

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