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{{Short description|Mathematical series}}

In [[mathematics]], the '''binomial series''' is a generalization of the polynomial that comes from a [[binomial formula]] expression like <math>(1+x)^n</math> for a nonnegative integer <math>n</math>.  Specifically, the binomial series is the [[MacLaurin series]] for the [[function (mathematics)|function]] <math>f(x)=(1+x)^{\alpha}</math>, where <math>\alpha \in \Complex</math> and <math>|x| < 1</math>. Explicitly,

{{NumBlk|:|<math>\begin{align}
 (1 + x)^\alpha  &= \sum_{k=0}^{\infty} \; \binom{\alpha}{k} \; x^k = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!} x^2 + \frac{\alpha(\alpha-1)(\alpha-2)}{3!} x^3 + \cdots
 \end{align}</math>|{{EquationRef|1}}}}

where the [[power series]] on the right-hand side of ({{EquationNote|1}}) is expressed in terms of the [[Binomial coefficient#Generalization and connection to the binomial series|(generalized) binomial coefficient]]s

:<math>\binom{\alpha}{k} := \frac{\alpha (\alpha-1) (\alpha-2) \cdots (\alpha-k+1)}{k!}. </math>

Note that if {{mvar|α}} is a nonnegative [[integer]] {{mvar|n}} then the {{math|''x''{{sup|''n'' + 1}}}} term and all later terms in the series are {{math|0}}, since each contains a factor of {{math|(''n'' − ''n'')}}. Thus, in this case, the series is finite and gives the algebraic [[binomial theorem|binomial formula]].

== Convergence ==
=== Conditions for convergence ===

Whether ({{EquationNote|1}}) [[convergent series|converges]] depends on the values of the complex numbers {{mvar|α}} and&nbsp;{{mvar|x}}. More precisely:

#If {{math|{{!}}''x''{{!}} < 1}}, the series converges [[absolute convergence|absolutely]] for any complex number {{mvar|α}}. 
#If {{math|1={{!}}''x''{{!}} = 1}}, the series converges absolutely [[if and only if]] either {{math|Re(''α'') > 0}} or {{math|1=''α'' = 0}}, where {{math|Re(''α'')}} denotes the [[complex number|real part]] of {{mvar|α}}.
# If {{math|1={{!}}''x''{{!}} = 1}} and {{math|''x'' ≠ −1}}, the series converges if and only if {{math|Re(''α'') > −1}}.
#If {{math|1=''x'' = −1}}, the series converges if and only if either {{math|Re(''α'') > 0}} or {{math|1=''α'' = 0}}.
#If {{math|{{!}}''x''{{!}} > 1}}, the series [[divergent series|diverges]] except when {{mvar|α}} is a non-negative integer, in which case the series is a finite sum.

In particular, if {{mvar|α}} is not a non-negative integer, the situation at the boundary of the [[radius of convergence|disk of convergence]], {{math|1={{abs|''x''}} = 1}}, is summarized as follows:

* If {{math|Re(''α'') > 0}}, the series converges absolutely.
* If {{math|−1 < Re(''α'') ≤ 0}}, the series converges [[conditional convergence|conditionally]] if {{math|''x'' ≠ −1}} and diverges if {{math|1=''x'' = −1}}.
* If {{math|Re(''α'') ≤ −1}}, the series diverges.

=== Identities to be used in the proof ===

The following hold for any complex number&nbsp;{{mvar|α}}:

:<math>{\alpha \choose 0} = 1,</math>
{{NumBlk|:|<math> {\alpha \choose k+1} = {\alpha\choose k}\,\frac{\alpha-k}{k+1},  </math>|{{EquationRef|2}}}}

{{NumBlk|:|<math> {\alpha \choose k-1} +  {\alpha\choose k} = {\alpha+1 \choose k}. </math>|{{EquationRef|3}}}}
Unless <math>\alpha</math> is a nonnegative integer (in which case the binomial coefficients vanish as <math>k</math> is larger than <math>\alpha</math>), a useful [[asymptotic analysis|asymptotic]] relationship for the binomial coefficients is, in [[Landau notation]]:

{{NumBlk|:|<math> {\alpha \choose k} = \frac{(-1)^k} {\Gamma(-\alpha)k^ {1+\alpha} } \,(1+o(1)), \quad\text{as }k\to\infty. </math>|{{EquationRef|4}}}}

This is essentially equivalent to Euler's definition of the [[Gamma function]]:

:<math>\Gamma(z) = \lim_{k \to \infty} \frac{k! \; k^z}{z \; (z+1)\cdots(z+k)}, </math>

and implies immediately the coarser bounds

{{NumBlk|:|<math> \frac {m} {k^{1+\operatorname{Re}\,\alpha}}\le \left|{\alpha \choose k}\right| \le \frac {M} {k^{1+\operatorname{Re}\alpha}}, </math>|{{EquationRef|5}}}}
for some positive constants {{mvar|m}} and {{mvar|M}} .

Formula ({{EquationNote|2}}) for the generalized binomial coefficient can be rewritten as
{{NumBlk|:|<math> {\alpha \choose  k} = \prod_{j=1}^k \left (\frac {\alpha + 1} j - 1 \right ). </math>|{{EquationRef|6}}}}

=== Proof ===

To prove (i) and (v), apply the [[ratio test]] and use formula ({{EquationNote|2}}) above to show that whenever <math>\alpha</math> is not a nonnegative integer, the [[radius of convergence]] is exactly&nbsp;1.  Part (ii) follows from formula ({{EquationNote|5}}), by comparison with the [[Convergence tests#p-series test|{{mvar|p}}-series]]

:<math> \sum_{k=1}^\infty \; \frac {1} {k^p}, </math>

with <math>p=1+\operatorname{Re}\alpha</math>. To prove (iii), first use formula ({{EquationNote|3}}) to obtain

{{NumBlk|:|<math>(1 + x) \sum_{k=0}^n \; {\alpha \choose k} \; x^k =\sum_{k=0}^n \; {\alpha+1\choose k} \; x^k + {\alpha \choose n} \;x^{n+1}, </math>|{{EquationRef|7}}}}

and then use (ii) and formula ({{EquationNote|5}}) again to prove convergence of the right-hand side when <math> \operatorname{Re} \alpha> - 1 </math> is assumed. On the other hand, the series does not converge if <math>|x|=1</math> and <math> \operatorname{Re} \alpha \le - 1 </math>, again by formula ({{EquationNote|5}}). Alternatively, we may observe that for all <math>j</math>, <math display="inline"> \left |\frac {\alpha + 1}{j} - 1 \right | \ge 1 - \frac {\operatorname{Re} \alpha + 1}{j} \ge 1 </math>. Thus, by formula ({{EquationNote|6}}), for all <math display="inline"> k, \left|{\alpha \choose  k} \right| \ge 1 </math>. This completes the proof of (iii). Turning to (iv), we use identity ({{EquationNote|7}}) above with <math>x=-1</math> and <math>\alpha-1</math> in place of <math>\alpha</math>, along with formula ({{EquationNote|4}}), to obtain

:<math>\sum_{k=0}^n \; {\alpha\choose k} \; (-1)^k = {\alpha-1 \choose n} \;(-1)^n= \frac{1} {\Gamma(-\alpha+1)n^{\alpha}}(1+o(1))</math>

as <math>n\to\infty</math>. Assertion (iv) now follows from the asymptotic behavior of the sequence <math>n^{-\alpha} = e^{-\alpha \log(n)}</math>. (Precisely, <math> \left|e^{-\alpha\log n}\right| = e^{-\operatorname{Re}\alpha\, \log n}</math>
certainly converges to <math>0</math> if <math>\operatorname{Re}\alpha>0</math> and diverges to <math>+\infty</math> if <math>\operatorname{Re}\alpha<0</math>. If <math>\operatorname{Re}\alpha=0</math>, then <math>n^{-\alpha} = e^{-i \operatorname{Im}\alpha\log n}</math> converges if and only if the sequence <math> \operatorname{Im}\alpha\log n </math> converges <math>\bmod{2\pi}</math>, which is certainly true if <math>\alpha=0</math> but false if <math>\operatorname{Im}\alpha \neq0</math>: in the latter case the sequence is dense <math>\bmod{2\pi}</math>, due to the fact that <math>\log n</math> diverges and <math>\log (n+1)-\log n</math> converges to zero).

== Summation of the binomial series ==

The usual argument to compute the sum of the binomial series goes as follows. [[Derivative|Differentiating]] term-wise the binomial series within the disk of convergence {{math|{{abs|''x''}} < 1}} and using formula ({{EquationNote|1}}), one has that the sum of the series is an [[analytic function]] solving the [[ordinary differential equation]] {{math|1=(1 + ''x'')''u''′(''x'') − ''αu''(''x'') = 0}} with [[initial condition]] {{math|1=''u''(0) = 1}}. 

The unique solution of this problem is the function {{math|1=''u''(''x'') = (1 + ''x'')<sup>''α''</sup>}}. Indeed, multiplying by the [[integrating factor]] {{math|1= (1 + ''x'')<sup>−''α''−1</sup>}} gives

:<math>0=(1+x)^{-\alpha}u'(x) - \alpha (1+x)^{-\alpha-1} u(x)=  \big[(1+x)^{-\alpha}u(x)\big]'\,,</math>

so the function {{math|1= (1 + ''x'')<sup>''−α''</sup>''u''(''x'')}} is a constant, which the initial condition tells us is {{math|1}}.  That is, {{math|1=''u''(''x'') = (1 + ''x'')<sup>''α''</sup>}} is the sum of the binomial series for {{math|{{abs|''x''}} < 1}}.

The equality extends to {{math|1={{abs|''x''}} = 1}} whenever the series converges, as a consequence of [[Abel's theorem]] and by [[continuous function|continuity]] of {{math|(1 + ''x'')<sup>''α''</sup>}}.

== Negative binomial series ==
Closely related is the ''negative binomial series'' defined by the [[MacLaurin series]] for the function <math>g(x)=(1-x)^{-\alpha}</math>, where <math>\alpha \in \Complex</math> and <math>|x| < 1</math>. Explicitly,
:<math>\begin{align}
\frac{1}{(1 - x)^\alpha} &= \sum_{k=0}^{\infty} \; \frac{g^{(k)}(0)}{k!} \; x^k \\
 &= 1 + \alpha x + \frac{\alpha(\alpha+1)}{2!} x^2 + \frac{\alpha(\alpha+1)(\alpha+2)}{3!} x^3 + \cdots,
 \end{align}</math>
which is written in terms of the [[multiset coefficient]]
:<math>\left(\!\!{\alpha\choose k}\!\!\right) := {\alpha+k-1 \choose k} = \frac{\alpha (\alpha+1) (\alpha+2) \cdots (\alpha+k-1)}{k!}\,.</math>

When {{mvar|α}} is a positive integer, several common sequences are apparent.  The case {{math|1=''α'' = 1}} gives the series {{math|1 + ''x'' + ''x''{{sup|2}} + ''x''{{sup|3}} + ...}}, where the coefficient of each term of the series is simply {{math|1}}.  The case {{math|1=''α'' = 2}} gives the series {{math|1 + 2''x'' + 3''x''{{sup|2}} + 4''x''{{sup|3}} + ...}}, which has the counting numbers as coefficients.  The case {{math|1=''α'' = 3}} gives the series {{math|1 + 3''x'' + 6''x''{{sup|2}} + 10''x''{{sup|3}} + ...}}, which has the [[triangle numbers]] as coefficients.  The case {{math|1=''α'' = 4}} gives the series {{math|1 + 4''x'' + 10''x''{{sup|2}} + 20''x''{{sup|3}} + ...}}, which has the [[tetrahedral numbers]] as coefficients, and similarly for higher integer values of {{mvar|α}}.

The negative binomial series includes the case of the [[geometric series]], the [[power series]]<ref>{{citation|url=https://maa.org/sites/default/files/0002989008055.di011924.01p0005u.pdf|title=The geometric series in calculus|author=[[George Andrews (mathematician)|George Andrews]]|journal= The American Mathematical Monthly|volume=105|issue=1|pages=36–40|doi=10.1080/00029890.1998.12004846|year=2018}}</ref>
<math display=block>\frac{1}{1-x} = \sum_{n=0}^\infty x^n</math> 
(which is the negative binomial series when <math>\alpha=1</math>, convergent in the disc <math>|x|<1</math>) and, more generally, series obtained by differentiation of the geometric power series:
<math display="block">\frac{1}{(1-x)^n} = \frac{1}{(n-1)!}\frac{d^{n-1}}{dx^{n-1}}\frac{1}{1-x}</math>
with <math>\alpha=n</math>, a positive integer.<ref>{{citation|first=Konrad|last=Knopp|authorlink=Konrad Knopp|title=Theory and applications of infinite series|year=1944|publisher=Blackie and Son}}, §22.</ref>

== History ==

The first results concerning binomial series for other than positive-integer exponents were given by Sir [[Isaac Newton]] in the study of [[area]]s enclosed under certain curves. [[John Wallis]] built upon this work by considering expressions of the form {{math|1=''y'' = (1 − ''x''<sup>2</sup>)<sup>''m''</sup>}} where {{mvar|m}} is a fraction. He found that (written in modern terms) the successive coefficients {{math|''c''<sub>''k''</sub>}} of {{math|(−''x''<sup>2</sup>)<sup>''k''</sup>}} are to be found by multiplying the preceding coefficient by {{sfrac|{{mvar|m}} &minus; ({{mvar|k}} &minus; 1)|{{mvar|k}}}} (as in the case of integer exponents), thereby implicitly giving a formula for these coefficients. He explicitly writes the following instances{{efn|{{sfn|Coolidge|1949}} In fact this source gives all non-constant terms with a negative sign, which is not correct for the second equation; one must assume this is an error of transcription.}}

:<math>(1-x^2)^{1/2}=1-\frac{x^2}2-\frac{x^4}8-\frac{x^6}{16}\cdots</math>

:<math>(1-x^2)^{3/2}=1-\frac{3x^2}2+\frac{3x^4}8+\frac{x^6}{16}\cdots</math>

:<math>(1-x^2)^{1/3}=1-\frac{x^2}3-\frac{x^4}9-\frac{5x^6}{81}\cdots</math>

The binomial series is therefore sometimes referred to as [[Binomial theorem#Newton's generalized binomial theorem|Newton's binomial theorem]]. Newton gives no proof and is not explicit about the nature of the series. Later, on 1826 [[Niels Henrik Abel]] discussed the subject in a paper published on ''[[Crelle's Journal]]'', treating notably questions of convergence.{{sfn|Abel|1826}}

==See also==
{{Portal|Mathematics}}
*[[Binomial approximation]]
*[[Binomial theorem#Newton's generalized binomial theorem|Binomial theorem]]
*[[Table of Newtonian series]]
*[[Lambert W function]]

== Footnotes ==
=== Notes ===
{{notelist}}

=== Citations ===
{{reflist|20em}}

== References ==
*{{Citation
|first=Niels
|last=Abel
|author-link=Niels Henrik Abel
|title=Recherches sur la série 1 + (''m''/1)x + (''m''(''m'' − 1)/1.2)x{{sup|2}} + (''m''(''m'' − 1)(''m'' − 2)/1.2.3)''x''{{sup|3}} + ...
|journal=[[Crelle's Journal|Journal für die reine und angewandte Mathematik]]
|volume=1
|issue=
|year=1826
|pages=311–339
|id=
|url=http://www.bibnum.education.fr/math%C3%A9matiques/alg%C3%A9bre/niels-abel-et-les-crit%C3%A8res-de-convergence
}}
* {{citation |jstor=2305028 |title= The Story of the Binomial Theorem| first= J. L. |last= Coolidge |journal= The American Mathematical Monthly | volume=56 | issue=3 |pages=147–157 |year=1949 |doi=10.2307/2305028 }}

== External links ==
*{{MathWorld | urlname= BinomialSeries | title= Binomial Series}}
*{{MathWorld | urlname= BinomialTheorem | title= Binomial Theorem}}
*{{PlanetMath | urlname= binomialformula | title= binomial formula}}
*{{SpringerEOM | author-first= E.D.|author-last= Solomentsev | title= Binomial series}}
*{{cite web|url=https://www.quantamagazine.org/how-isaac-newton-discovered-the-binomial-power-series-20220831/|title=How Isaac Newton Discovered the Binomial Power Series|date=August 31, 2022}}

{{Calculus topics}}

[[Category:Complex analysis]]
[[Category:Factorial and binomial topics]]
[[Category:Mathematical series]]
[[Category:Real analysis]]