Binomial Theorem for Negative Index and Negative Parameter

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Theorem

Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $z \in \R$ be a real number such that $\size z < 1$.


Then:

\(\displaystyle \dfrac 1 {\paren {1 - z}^{n + 1} }\) \(=\) \(\displaystyle \sum_{k \mathop \ge 0} \binom {-n - 1} k \paren {-z}^k\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \ge 0} \binom {n + k} n z^k\)

where $\dbinom {n + k} n$ denotes a binomial coefficient.


Proof

\(\displaystyle \dfrac 1 {\paren {1 - z}^{n + 1} }\) \(=\) \(\displaystyle \paren {1 + \paren {-z} }^{- n - 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \ge 0} \binom {- n - 1} k \paren {-z}^k\) General Binomial Theorem
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \ge 0} \dbinom {n + 1 + k - 1} k \paren {-1}^k \paren {-z}^k\) Negated Upper Index of Binomial Coefficient
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \ge 0} \dbinom {n + k} k z^k\)

$\blacksquare$


Sources