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| Title of object: |
binomial formula |
| Canonical Name: |
BinomialFormula |
| Type: |
Theorem |
| Created on: |
2002-02-19 12:43:45 |
| Modified on: |
2005-07-11 07:29:24 |
| Classification: |
msc:26A06 |
Preamble:
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\natnums}{\mathbb{N}}
\newcommand{\cnums}{\mathbb{C}}
\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\lb}{\left[}
\newcommand{\rb}{\right]}
\newcommand{\supth}{^{\text{th}}}
\newtheorem{proposition}{Proposition} |
Content:
The binomial formula gives the power series expansion of the
$p\supth$ power function for every real power $p$. To wit,
$$(1+x)^p = \sum_{n=0}^\infty p^{\underline{n}} \, \frac{x^n}{n!},\quad
x\in\reals,\;|x|<1,$$
where
$$p^{\underline{n}}= p(p-1)\ldots (p-n+1)$$
denotes the $n\supth$
falling factorial of $p$.
Note that for $p\in\natnums$ the power series reduces to a
polynomial. The above formula is therefore a generalization of the
binomial theorem.
Also note that the binomial formula is valid at $x=\pm 1$, but for certain values of $p$ only. Of course, we have convergence if $p$ is a natural number. Furthermore, for $x=1$, we have absolute convergence if $p>0$, and conditional convergence if $-1<p<0$. For $x=-1$ we have absolute convergence for $p>0$. |
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