Binomial Series Expander — Theory
1. Ordinary binomial theorem
For a nonnegative integer \(m\), the binomial theorem says
\[
(1+x)^m
=
\sum_{n=0}^{m}\binom{m}{n}x^n.
\]
This is a finite polynomial because the coefficients become zero after \(n=m\).
2. Generalized binomial theorem
The generalized binomial theorem extends the idea to exponents that do not have to be nonnegative integers.
\[
(1+x)^\alpha
=
\sum_{n=0}^{\infty}\binom{\alpha}{n}x^n.
\]
Here \(\alpha\) can be fractional, negative, or decimal.
3. Generalized binomial coefficient
The coefficient is defined by
\[
\binom{\alpha}{n}
=
\frac{\alpha(\alpha-1)(\alpha-2)\cdots(\alpha-n+1)}{n!}.
\]
For \(n=0\), the coefficient is
\[
\binom{\alpha}{0}=1.
\]
4. Recurrence for coefficients
After finding one coefficient, the next coefficient can be found quickly:
\[
\binom{\alpha}{n}
=
\binom{\alpha}{n-1}
\frac{\alpha-n+1}{n}.
\]
This is useful for building the table term by term.
5. Partial binomial polynomial
The order-\(N\) binomial polynomial is
\[
P_N(x)
=
\sum_{n=0}^{N}\binom{\alpha}{n}x^n.
\]
It approximates \((1+x)^\alpha\) inside the convergence interval.
6. Example: \((1+x)^{1/2}\)
Let \(\alpha=\frac12\). The first coefficients are found as follows:
\[
\binom{1/2}{0}=1.
\]
\[
\binom{1/2}{1}
=
\frac{1/2}{1}
=
\frac12.
\]
\[
\binom{1/2}{2}
=
\frac{(1/2)(1/2-1)}{2!}
=
\frac{(1/2)(-1/2)}{2}
=
-\frac18.
\]
\[
\binom{1/2}{3}
=
\frac{(1/2)(-1/2)(-3/2)}{3!}
=
\frac{1}{16}.
\]
Therefore,
\[
(1+x)^{1/2}
=
1+\frac12x-\frac18x^2+\frac1{16}x^3-\cdots.
\]
7. Convergence interval
For noninteger \(\alpha\), the main convergence region is
\[
|x|<1.
\]
Endpoint behavior depends on \(\alpha\).
\[
\begin{aligned}
x=1&:\ \text{converges if }\alpha>-1,\\
x=-1&:\ \text{converges if }\alpha>0.
\end{aligned}
\]
8. Integer exponents
If \(\alpha\) is a nonnegative integer, the binomial series terminates.
For example,
\[
(1+x)^3
=
1+3x+3x^2+x^3.
\]
In this case, the expression is an ordinary polynomial and is valid for all real \(x\).
9. Negative exponents
Negative exponents give infinite series. For example,
\[
(1+x)^{-1}
=
1-x+x^2-x^3+\cdots,
\qquad |x|<1.
\]
This is the geometric series with ratio \(-x\).
10. Approximation error
At a chosen point \(x_0\), the approximation error is
\[
\text{Error}
=
(1+x_0)^\alpha-P_N(x_0).
\]
Inside the convergence interval, the error usually decreases as \(N\) increases.
11. What the graph shows
The graph compares two curves:
\[
y=(1+x)^\alpha
\qquad
\text{and}
\qquad
y=P_N(x).
\]
The shaded region marks the main convergence window \(-1
13. Common mistakes
- Forgetting the decreasing factors: the coefficient uses \(\alpha,\alpha-1,\alpha-2,\ldots\), not powers of \(\alpha\).
- Forgetting \(n!\): every coefficient is divided by \(n!\).
- Using the expansion outside its convergence interval: for noninteger \(\alpha\), the main interval is \(|x|<1\).
- Confusing finite and infinite cases: nonnegative integer exponents give finite polynomials.
- Ignoring endpoint behavior: \(x=1\) and \(x=-1\) require separate checks.
