# Binomial series
The binomial series is the power series
\[ (1+z)^{\alpha} = \sum_{n=0}^{\infty} \binom{\alpha}{n} z^{n} = 1 + \alpha z + \frac{\alpha(\alpha-1)}{2!} z^{2} + \frac{\alpha(\alpha-1)(\alpha-2)}{3!} z^{3} + \cdots, \]
in which \( \alpha \) is an arbitrary fixed number, in general complex, and the generalized binomial coefficients are defined by \( \binom{\alpha}{0} = 1 \) and \( \binom{\alpha}{n} = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!} \) for \( n \ge 1 \)[1](https://encyclopediaofmath.org/wiki/Binomial_series) • [2](https://www.math.ualberta.ca/~isaac/math311/s14/binomial_series.pdf). When \( \alpha \) is a nonnegative integer the series terminates and becomes the ordinary finite binomial theorem; for every other value of \( \alpha \) it is an infinite series whose convergence must be analyzed[1](https://encyclopediaofmath.org/wiki/Binomial_series).
| Key fact | Detail |
|---|---|
| Definition | \( \sum_{n=0}^{\infty} \binom{\alpha}{n} z^{n} \) with \( \binom{\alpha}{n} = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!} \), for any complex \( \alpha \)[1](https://encyclopediaofmath.org/wiki/Binomial_series) |
| Finite case | For \( \alpha = m \ge 0 \) an integer, the series reduces to \( m+1 \) terms, the Newton binomial \( (1+z)^m \)[1](https://encyclopediaofmath.org/wiki/Binomial_series) |
| Radius of convergence | \( R = 1 \): absolute convergence for \( \lvert z \rvert < 1 \), divergence for \( \lvert z \rvert > 1 \)[2](https://www.math.ualberta.ca/~isaac/math311/s14/binomial_series.pdf) |
| Boundary, \( \operatorname{Re} \alpha > 0 \) | Converges absolutely at every point of \( \lvert z \rvert = 1 \)[1](https://encyclopediaofmath.org/wiki/Binomial_series) |
| Boundary, \( -1 < \operatorname{Re} \alpha \le 0 \) | Diverges at \( z = -1 \), converges conditionally at all other points of the unit circle[1](https://encyclopediaofmath.org/wiki/Binomial_series) |
| Boundary, \( \operatorname{Re} \alpha \le -1 \) | Diverges at every point of \( \lvert z \rvert = 1 \)[1](https://encyclopediaofmath.org/wiki/Binomial_series) |
| Sum | At every point of convergence the series equals the principal value of \( (1+z)^{\alpha} \), normalized to 1 at \( z = 0 \)[1](https://encyclopediaofmath.org/wiki/Binomial_series) |
| History | Probably first mentioned by Newton in 1664–1665; studied exhaustively by Abel, work that started the theory of complex power series[1](https://encyclopediaofmath.org/wiki/Binomial_series) |
## Convergence
Inside the unit disk the behavior is uniform across all values of \( \alpha \) that are not nonnegative integers: the series converges absolutely when \( \lvert z \rvert < 1 \) and diverges when \( \lvert z \rvert > 1 \), so the radius of convergence is \( R = 1 \)[2](https://www.math.ualberta.ca/~isaac/math311/s14/binomial_series.pdf). The radius can be read from the ratio test applied to the coefficients, and it has a geometric meaning: a power series about a point extends to the nearest point where the function fails to be analytic, and \( (1+z)^{\alpha} \) for noninteger \( \alpha \) has a branch point (point where a function becomes multi-valued or singular) at \( z = -1 \), exactly one unit from the center of expansion[3](https://books.physics.oregonstate.edu/GMM/convergence.html).
**The unit circle requires separate analysis.** The general theorem for power series says nothing about the boundary circle, where a series may converge absolutely, converge conditionally, or diverge[4](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_15.html). For the binomial series the answer depends on the real part of \( \alpha \)[1](https://encyclopediaofmath.org/wiki/Binomial_series):
1. If \( \operatorname{Re} \alpha > 0 \), the series converges absolutely at all points with \( \lvert z \rvert = 1 \).
2. If \( \operatorname{Re} \alpha \le -1 \), it diverges at all points with \( \lvert z \rvert = 1 \).
3. If \( -1 < \operatorname{Re} \alpha \le 0 \), it diverges at \( z = -1 \) and converges conditionally at every other point of the unit circle.
The Alberta notes add a uniform-convergence statement for the intermediate case: when \( -1 < \operatorname{Re} \alpha \le 0 \), the series converges uniformly on the set \( \{ z : \lvert z \rvert \le 1, \ \lvert 1+z \rvert \ge \delta \} \) for any \( 0 < \delta < 2 \), that is, on the closed unit disk with a neighborhood of the bad point \( z = -1 \) removed[2](https://www.math.ualberta.ca/~isaac/math311/s14/binomial_series.pdf).
**Real variables.** When \( z = x \) and \( \alpha \) are real and \( \alpha \) is not a nonnegative integer, the picture is[1](https://encyclopediaofmath.org/wiki/Binomial_series):
- \( \alpha > 0 \): absolute convergence on the whole closed interval \( -1 \le x \le 1 \).
- \( \alpha \le -1 \): absolute convergence only for \( -1 < x < 1 \), divergence at both endpoints and beyond.
- \( -1 < \alpha \le 0 \): absolute convergence for \( -1 < x < 1 \), conditional convergence at \( x = 1 \), divergence at \( x = -1 \).
- For \( \lvert x \rvert > 1 \) the series always diverges.
Calculus texts usually state only the interior guarantee, that \( (1+x)^k = \sum_{n=0}^{\infty} \binom{k}{n} x^n \) for \( \lvert x \rvert < 1 \) with any real \( k \), and leave the endpoints to be checked case by case[5](https://tutorial.math.lamar.edu/classes/calcii/binomialseries.aspx).
## The sum of the series
Wherever it converges, the binomial series represents the principal value of \( (1+z)^{\alpha} \), defined as \( e^{\alpha \operatorname{Log}(1+z)} \) for \( z \ne -1 \) using the principal branch of the logarithm, with the value at \( z = 0 \) equal to 1[1](https://encyclopediaofmath.org/wiki/Binomial_series) • [2](https://www.math.ualberta.ca/~isaac/math311/s14/binomial_series.pdf). The branch point at \( z = -1 \) is what caps the radius of convergence at 1 no matter how smooth the function looks on the real line[3](https://books.physics.oregonstate.edu/GMM/convergence.html).
The series is a special case of a [hypergeometric series](https://www.edgepedia.org/hypergeometric-series)[1](https://encyclopediaofmath.org/wiki/Binomial_series), which places it inside the larger family of classical special-function expansions.
## Use in expanding functions
Any expression can be put in binomial form by factoring so that the variable part is small. For example, to expand \( \sqrt{9-x} \), write \( 9 - x = 9\left(1 - \frac{x}{9}\right) \), so that with \( k = \tfrac{1}{2} \) and \( x \) replaced by \( -x/9 \),
\[ \sqrt{9-x} = 3\left(1 - \frac{x}{9}\right)^{1/2} = 3 - \frac{x}{6} - \frac{x^2}{216} - \frac{x^3}{3888} - \cdots, \]
valid while \( \lvert x/9 \rvert < 1 \)[5](https://tutorial.math.lamar.edu/classes/calcii/binomialseries.aspx). The binomial expansion also generates series for related functions; for instance, expanding \( 1/(z^2+1) \) about \( z = 0 \) with the binomial expansion yields a series valid for \( \lvert z \rvert \le 1 \), limited by the complex poles at \( z = \pm i \)[3](https://books.physics.oregonstate.edu/GMM/convergence.html).
## History
**Newton and fractional powers.** The binomial series was probably first mentioned by [Isaac Newton](https://www.edgepedia.org/isaac-newton) in 1664–1665[1](https://encyclopediaofmath.org/wiki/Binomial_series). According to the historian Niccolò Guicciardini, professor of history of science at the [University of Bologna](https://www.edgepedia.org/university-of-bologna), Newton's method with infinite series was heavily derived from John Wallis, Savilian professor of geometry at Oxford, whose *Arithmetica Infinitorum* dealt with the quadrature of curvilinear figures. Using a Wallisian interpolation technique, Newton arrived at the binomial series of fractional powers, and this discovery gave him access to the realm of infinite series found in his work *De Analysi per aequationes numero terminorum infinitas*[6](https://doi.org/10.7551/mitpress/9780262013178.003.0007). In the summer of 1669, Isaac Barrow and John Collins, a mathematical intelligencer, spread Newton's treatise on analyzing equations through infinite series among a circle of London mathematicians; circulated in manuscript rather than print, it amounts to his first publication.[7](https://dev-newtonproject.history.ox.ac.uk/texts/introduction) His work on analysis with infinite series was written in 1669 but not published until 1711[8](https://mathshistory.st-andrews.ac.uk/HistTopics/The_rise_of_calculus/).
**Abel and rigor.** An exhaustive study of the binomial series was conducted by [Niels Henrik Abel](https://www.edgepedia.org/niels-henrik-abel), whose 1826 paper "Untersuchungen über die Reihe \( 1 + mx + \frac{m(m-1)x^2}{2 \cdot 1} + \cdots \)" in *Journal für die reine und angewandte Mathematik* treated its convergence in full; this work was the starting point of the theory of complex power series[1](https://encyclopediaofmath.org/wiki/Binomial_series). The boundary cases tabulated above are exactly the kind of result this theory made possible: three series can share the same radius of convergence yet behave differently on the boundary circle, converging absolutely, converging conditionally, or diverging[4](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_15.html).
## References
1. [Binomial series, Encyclopedia of Mathematics (E.D. Solomentsev, originator)](https://encyclopediaofmath.org/wiki/Binomial_series)
2. [The Binomial Series, Math 311 Theory of Functions of a Complex Variable, University of Alberta](https://www.math.ualberta.ca/~isaac/math311/s14/binomial_series.pdf)
3. [Section 13.9 Convergence of Power Series, Oregon State University](https://books.physics.oregonstate.edu/GMM/convergence.html)
4. [Lecture 15: Power series, Complex Analysis, UC Berkeley](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_15.html)
5. [Calculus II: Binomial Series, Paul's Online Math Notes, Lamar University](https://tutorial.math.lamar.edu/classes/calcii/binomialseries.aspx)
6. [Niccolò Guicciardini, "The Method of Series," in *Isaac Newton on Mathematical Certainty and Method*, MIT Press, 2009](https://doi.org/10.7551/mitpress/9780262013178.003.0007)
7. [Introduction to the Texts, Newton Project, University of Oxford](https://dev-newtonproject.history.ox.ac.uk/texts/introduction)
8. [The rise of calculus, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/The_rise_of_calculus/)
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