Binomial SumsBinomial SeriesTrinomial Series

Section 12.2 Binomial SumsBinomial SeriesTrinomial Series

The binomial series is also foundational. It is technically not a series since the sum_if finite but we won’t bother with that for now. It is given by

$\begin{gather*} B = \sum_{k=0}^{n} {\binom{n}{k} a^k b^{n-k}} \end{gather*}$

Theorem 12.2.1 Binomial Theorem

For

$n \in \mathbb{N} \text{,}$

$\displaystyle {(a+b)^n = \sum_{k=0}^{n} {\binom{n}{k} a^k b^{n-k}}}$

Proof

By induction:

Basic Step: n = 1 is trivial

Inductive Step: Assume the statement is true as given for some $n \ge 1\text{.}$ Show $(a+b)^{n+1} = \sum_{k=0}^{n+1} {\binom{n+1}{k} a^k b^{n+1-k}}$

\begin{align*} (a+b)^{n+1} & = (a+b)(a+b)^n\\ & = (a+b)\sum_{k=0}^{n} {\binom{n}{k} a^k b^{n-k}}\\ & = \sum_{k=0}^n \binom{n}{k} a^{k+1} b^{n-k} + \sum_{k=0}^n \binom{n}{k} a^k b^{n-k+1}\\ & = \sum_{k=0}^{n-1} \binom{n}{k} a^{k+1} b^{n-k} + a^{n+1} + b^{n+1} + \sum_{k=1}^n \binom{n}{k} a^k b^{n-k+1}\\ & = \sum_{j=1}^n \binom{n}{j-1} a^j b^{n-(j-1)} + a^{n+1} + b^{n+1} + \sum_{k=1}^n \binom{n}{k} a^k b^{n+1-k}\\ & = b^{n+1} + \sum_{k=1}^n \left [ \binom{n}{k-1} + \binom{n}{k} \right ] a^k b^{n+1-k} + a^{n+1}\\ & = b^{n+1} + \sum_{k=1}^n \binom{n+1}{k} a^k b^{n+1-k} + a^{n+1}\\ & = \sum_{k=0}^{n+1} \binom{n+1 }{k} a^k b^{n+1-k} \end{align*}

Consider $B(a,b) = \sum_{k=0}^{n} {\binom{n}{k} a^k b^{n-k}}\text{.}$ This finite sum_is known as the Binomial Series.

Show that $B(a,b) = (a+b)^n$

Show that $B(1,1) = 2^n$

Show that $B(-1,1) = 0$

Show that $B(p,1-p) = 1$

Easily, $B(x,1) = \sum_{k=0}^{n} {\binom{n}{k} a^k}$

$\begin{gather*} (a+b+c)^n = \sum_{k_1+k_2+k_3=n}^{} {\binom{n}{k_1,k_2,k_3} a^{k_1} b^{k_2} c^{k_3}} \end{gather*}$

where $\binom{n}{k_1,k_2,k_3} = \frac{n!}{k_1!k_2!k_3!}\text{.}$ This can be generalized to any number of terms to give what is know as a multinomial series.