Summation proof
09-13-2023, 03:52 AM
Post: #5
 Albert Chan Senior Member Posts: 2,680 Joined: Jul 2018
RE: Summation proof
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.
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 Messages In This Thread Summation proof - Albert Chan - 09-13-2023, 02:11 AM RE: Summation proof - rprosperi - 09-13-2023, 02:37 AM RE: Summation proof - Albert Chan - 09-13-2023, 03:20 AM RE: Summation proof - Albert Chan - 09-13-2023, 02:54 AM RE: Summation proof - Albert Chan - 09-13-2023 03:52 AM RE: Summation proof - John Keith - 09-13-2023, 01:44 PM RE: Summation proof - Albert Chan - 09-13-2023, 07:12 PM RE: Summation proof - rprosperi - 09-13-2023, 06:49 PM RE: Summation proof - John Keith - 09-13-2023, 08:15 PM RE: Summation proof - Maximilian Hohmann - 09-13-2023, 08:35 PM RE: Summation proof - Albert Chan - 09-13-2023, 11:18 PM RE: Summation proof - rprosperi - 09-14-2023, 11:53 AM

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