Summation proof

[Albert Chan]

Another way to do this, very simply, without knowing Psi, Gamma ...

Product rule: (u * v)' = u * v' + v * u'

(09-13-2023 03:20 AM)Albert Chan Wrote:  Eureka! RHS must be \(\displaystyle \frac{d}{dn} \binom{n}{3}\) in disguise.

\(\displaystyle RHS = \frac{d}{dn} \frac{(n)(n\!-\!1)(n\!-\!2)}{3!}
= \frac{(n\!-\!1)(n\!-\!2) + n(n\!-\!2) + n(n\!-\!1)}{3!}
= \binom{n}{3} \left( \frac{1}{n} + \frac{1}{n\!-\!1} + \frac{1}{n\!-\!2}\right)
= LHS
\)

Harmonic series appears because product rule act on 1 factor at time.