Spencer, I am skeptical about your formula.
you posted:
$ \sum_{i=1}^n\frac{n}{n - i + 1}\! $
being equal to
$ \sum_{i=1}^n a_i = \frac{n(a_1+a_n)}{2}\! $
Try n=3
original expected value would be 11/2 = 5 1/2
your formula yields 6.
My calculation my be wrong, but otherwise I am thinking I might leave the formula as the answer.
