Binomial Theorem/Abel's Generalisation/Negative n

Theorem

$\ds \paren {x + y}^n = \sum_k \binom n k x \paren {x - k z}^{k - 1} \paren {y + k z}^{n - k}$

does not hold for $n \in \Z_{< 0}$.

Putting $n = x = -1$ and $y = z = 1$ into the left hand side

$\paren {-1 + 1}^{-1} = \dfrac 1 0$

which is undefined.

Putting the same values into the right hand side gives:

$\ds $

$\ds \sum_k \dbinom {-1} k \paren {-1} \paren {-1 - k}^{k - 1} \paren {1 + k}^{-1 - k}$

$\ds $

$=$

$\ds \sum_k \paren {-1}^k \dbinom {-1} k \paren {1 + k}^{-2}$

$\ds $

$=$

$\ds \sum_{k \mathop \ge 0} \paren {1 + k}^{-2}$

$\ds $

$=$

$\ds \sum_{k \mathop \ge 1} \dfrac 1 {k^2}$

$\ds $

$=$

$\ds \dfrac {\pi^2} 6$

$\blacksquare$