Simplify $\sum_{i=1}^n\frac{i-1}{(n-i+1)^2}$ to $\sum_{i=1}^n\frac{n-i}{i^2}$

Question

I'm trying to derive the variance of the coupon collector's problem. I'm following these lecture notes on page 2 http://www.cs.utexas.edu/~ecprice/courses/randomized/notes/lec5.pdf and trying the algebra but I can't simplify $\sum_{i=1}^n\frac{i-1}{(n-i+1)^2}$ to $\sum_{i=1}^n\frac{n-i}{i^2}$. Is there an algebra trick I am not noticing? Thanks!