| Version 7 |
Version 6 |
| The binomial formula gives the power series expansion of the |
The binomial formula gives the power series expansion of the |
| $p\supth$ power function. The power $p$ can be an integer, |
$p\supth$ power function. The power $p$ can be an integer, |
| rational, real, or even a complex number. The formula is |
rational, real, or even a complex number. The formula is |
| \begin{align*} |
\begin{align*} |
| (1+x)^p &= \sum_{n=0}^\infty \frac{p^{\underline{n}}}{n!} \, x^n\\ |
(1+x)^p &= \sum_{n=0}^\infty \frac{p^{\underline{n}}}{n!} \, x^n\\ |
| &= \sum_{n=0}^\infty \binom{p}{n} x^n |
&= \sum_{n=0}^\infty \binom{p}{n} x^n |
| \end{align*} |
\end{align*} |
| where $p^{\underline{n}}= p(p-1)\ldots (p-n+1)$ denotes the falling |
where $p^{\underline{n}}= p(p-1)\ldots (p-n+1)$ denotes the falling |
| factorial, and where $\binom{p}{n}$ denotes the generalized binomial |
factorial, and where $\binom{p}{n}$ denotes the generalized binomial |
| coefficient. |
coefficient. |
|
|
| For $p=0,1,2,\ldots$ the power series reduces to a polynomial, and we |
For $p=0,1,2,\ldots$ the power series reduces to a polynomial, and we |
| obtain the usual binomial theorem. For other values of $p$, the |
obtain the usual binomial theorem. For other values of $p$, the |
| radius of convergence of the series is $1$; the right-hand series |
radius of convergence of the series is $1$; the right-hand series |
| converges pointwise for all complex $|x|<1$ to the value on the left |
converges pointwise for all complex $|x|<1$ to the value on the left |
| side. Also note that the binomial formula is valid at $x=\pm 1$, but |
side. 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 $p$ |
for certain values of $p$ only. Of course, we have convergence if $p$ |
| is a natural number. Furthermore, for $x=1$ and real $p$, we have |
is a natural number. Furthermore, for $x=1$ and real $p$, we have |
| absolute convergence if $p>0$, and conditional convergence if |
absolute convergence if $p>0$, and conditional convergence if |
| $-1<p<0$. For $x=-1$ we have absolute convergence for $p>0$. |
$-1<p<0$. For $x=-1$ we have absolute convergence for $p>0$. |
