Inmathematics, thebinomial series is a generalization of thebinomial formula to cases where theexponent is not a positive integer:

1

where${\displaystyle \alpha }$ is anycomplex number, and thepower series on the right-hand side is expressed in terms of the(generalized) binomial coefficients

${\displaystyle (\genfrac{}{}{0ex}{}{\alpha }{k})=\frac{\alpha (\alpha -1)(\alpha -2)\cdots (\alpha -k+1)}{k!}.}$

The binomial series is theMacLaurin series for thefunction${\displaystyle f(x)=(1+x{)}^{\alpha }}$. It converges when.

Ifα is a nonnegativeintegern then thex^{n + 1} term and all later terms in the series are0, since each contains a factor of(n −n). In this case, the series is a finite polynomial, equivalent to the binomial formula.

Whether (1)converges depends on the values of the complex numbersα and x. More precisely:

1. If|x| < 1, the series convergesabsolutely for any complex numberα.
2. If|x| = 1, the series converges absolutelyif and only if eitherRe(α) > 0 orα = 0, whereRe(α) denotes thereal part ofα.
3. If|x| = 1 andx ≠ −1, the series converges if and only ifRe(α) > −1.
4. Ifx = −1, the series converges if and only if eitherRe(α) > 0 orα = 0.
5. If|x| > 1, the seriesdiverges except whenα is a non-negative integer, in which case the series is a finite sum.

In particular, ifα is not a non-negative integer, the situation at the boundary of thedisk of convergence,|x| = 1, is summarized as follows:

• IfRe(α) > 0, the series converges absolutely.
• If−1 < Re(α) ≤ 0, the series convergesconditionally ifx ≠ −1 and diverges ifx = −1.
• IfRe(α) ≤ −1, the series diverges.

The following hold for any complex number α:

${\displaystyle (\genfrac{}{}{0ex}{}{\alpha }{0})=1,}$

${\displaystyle (\genfrac{}{}{0ex}{}{\alpha }{k+1})=(\genfrac{}{}{0ex}{}{\alpha }{k})\frac{\alpha -k}{k+1},}$

2

${\displaystyle (\genfrac{}{}{0ex}{}{\alpha }{k-1})+(\genfrac{}{}{0ex}{}{\alpha }{k})=(\genfrac{}{}{0ex}{}{\alpha +1}{k}).}$

3

Unless${\displaystyle \alpha }$ is a nonnegative integer (in which case the binomial coefficients vanish as${\displaystyle k}$ is larger than${\displaystyle \alpha }$), a usefulasymptotic relationship for the binomial coefficients is, inLandau notation:

${\displaystyle (\genfrac{}{}{0ex}{}{\alpha }{k})=\frac{(-1{)}^k}{\mathrm{\Gamma }(-\alpha )k^{1+\alpha }}(1+o(1)),\text{as }k\to \mathrm{\infty }.}$

4

This is essentially equivalent to Euler's definition of theGamma function:

${\displaystyle \mathrm{\Gamma }(z)=\underset{k\to \mathrm{\infty }}{lim}\frac{k!k^z}{z(z+1)\cdots (z+k)},}$

and implies immediately the coarser bounds

${\displaystyle \frac{m}{k^{1+\mathrm{Re}\alpha }}\le \left|(\genfrac{}{}{0ex}{}{\alpha }{k})\right|\le \frac{M}{k^{1+\mathrm{Re}\alpha }},}$

5

for some positive constantsm andM .

Formula (2) for the generalized binomial coefficient can be rewritten as

${\displaystyle (\genfrac{}{}{0ex}{}{\alpha }{k})=\prod _{j=1}^k\left(\frac{\alpha +1}{j}-1\right).}$

6

To prove (i) and (v), apply theratio test and use formula (2) above to show that whenever${\displaystyle \alpha }$ is not a nonnegative integer, theradius of convergence is exactly 1. Part (ii) follows from formula (5), by comparison with thep-series

${\displaystyle \sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^p},}$

with${\displaystyle p=1+\mathrm{Re}(\alpha )}$. To prove (iii), first use formula (3) to obtain

${\displaystyle (1+x)\sum _{k=0}^n(\genfrac{}{}{0ex}{}{\alpha }{k})x^k=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{\alpha +1}{k})x^k+(\genfrac{}{}{0ex}{}{\alpha }{n})x^{n+1},}$

7

and then use (ii) and formula (5) again to prove convergence of the right-hand side when${\displaystyle \mathrm{Re}(\alpha )>-1}$ is assumed. On the other hand, the series does not converge if${\displaystyle |x|=1}$ and${\displaystyle \mathrm{Re}(\alpha )\le -1}$, again by formula (5). Alternatively, we may observe that for all${\displaystyle j}$,$\left|\frac{\alpha +1}{j}-1\right|\ge 1-\frac{\mathrm{Re}(\alpha )+1}{j}\ge 1$. Thus, by formula (6), for all$k,\left|(\genfrac{}{}{0ex}{}{\alpha }{k})\right|\ge 1$. This completes the proof of (iii). Turning to (iv), we use identity (7) above with${\displaystyle x=-1}$ and${\displaystyle \alpha -1}$ in place of${\displaystyle \alpha }$, along with formula (4), to obtain

${\displaystyle \sum _{k=0}^n(\genfrac{}{}{0ex}{}{\alpha }{k})(-1{)}^k=(\genfrac{}{}{0ex}{}{\alpha -1}{n})(-1{)}^n=\frac{1}{\mathrm{\Gamma }(-\alpha +1)n^{\alpha }}(1+o(1))}$

as${\displaystyle n\to \mathrm{\infty }}$. Assertion (iv) now follows from the asymptotic behavior of the sequence${\displaystyle n^{-\alpha }=e^{-\alpha \log (n)}}$. (Precisely,${\displaystyle \left|e^{-\alpha \log n}\right|=e^{-\mathrm{Re}(\alpha )\log n}}$certainly converges to${\displaystyle 0}$ if${\displaystyle \mathrm{Re}(\alpha )>0}$ and diverges to${\displaystyle +\mathrm{\infty }}$ if. If${\displaystyle \mathrm{Re}(\alpha )=0}$, then${\displaystyle n^{-\alpha }=e^{-i\mathrm{Im}(\alpha )\log n}}$ converges if and only if the sequence${\displaystyle \mathrm{Im}(\alpha )\log n}$ converges${\displaystyle mod2\pi }$, which is certainly true if${\displaystyle \alpha =0}$ but false if${\displaystyle \mathrm{Im}(\alpha )\ne 0}$: in the latter case the sequence is dense${\displaystyle mod2\pi }$, due to the fact that${\displaystyle \log n}$ diverges and${\displaystyle \log (n+1)-\log n}$ converges to zero).

The usual argument to compute the sum of the binomial series goes as follows.Differentiating term-wise the binomial series within the disk of convergence|x| < 1 and using formula (1), one has that the sum of the series is ananalytic function solving theordinary differential equation(1 +x)u′(x) −αu(x) = 0 withinitial conditionu(0) = 1.

The unique solution of this problem is the functionu(x) = (1 +x)^α. Indeed, multiplying by theintegrating factor(1 +x)^−α−1 gives

${\displaystyle 0=(1+x{)}^{-\alpha }{u}^{\prime }(x)-\alpha (1+x{)}^{-\alpha -1}u(x)=[(1+x{)}^{-\alpha }u(x){]}^{\prime },}$

so the function(1 +x)^−αu(x) is a constant, which the initial condition tells us is1. That is,u(x) = (1 +x)^α is the sum of the binomial series for|x| < 1.

The equality extends to|x| = 1 whenever the series converges, as a consequence ofAbel's theorem and bycontinuity of(1 +x)^α.

Closely related is thenegative binomial series defined by theMacLaurin series for the function${\displaystyle g(x)=(1-x{)}^{-\alpha }}$, where${\displaystyle \alpha \in \mathbb{C}}$ and. Explicitly,

which is written in terms of themultiset coefficient

${\displaystyle \left((\genfrac{}{}{0ex}{}{\alpha }{k})\right)=(\genfrac{}{}{0ex}{}{\alpha +k-1}{k})=\frac{\alpha (\alpha +1)(\alpha +2)\cdots (\alpha +k-1)}{k!}.}$

Whenα is a positive integer, several common sequences are apparent. The caseα = 1 gives the series1 +x +x² +x³ + ..., where the coefficient of each term of the series is simply1. The caseα = 2 gives the series1 + 2x + 3x² + 4x³ + ..., which has the counting numbers as coefficients. The caseα = 3 gives the series1 + 3x + 6x² + 10x³ + ..., which has thetriangle numbers as coefficients. The caseα = 4 gives the series1 + 4x + 10x² + 20x³ + ..., which has thetetrahedral numbers as coefficients, and similarly for higher integer values ofα.

The negative binomial series includes the case of thegeometric series, thepower series^[1]$${\displaystyle \frac{1}{1-x}=\sum _{n=0}^{\mathrm{\infty }}{x}^n}$$ (which is the negative binomial series when${\displaystyle \alpha =1}$, convergent in the disc) and, more generally, series obtained by differentiation of the geometric power series:$${\displaystyle \frac{1}{(1-x{)}^n}=\frac{1}{(n-1)!}\frac{d^{n-1}}{d{x}^{n-1}}\frac{1}{1-x}}$$with${\displaystyle \alpha =n}$, a positive integer.^[2]

The first results concerning binomial series for other than positive-integer exponents were given by SirIsaac Newton in the study ofareas enclosed under certain curves.John Wallis built upon this work by considering expressions of the formy = (1 −x²)^m wherem is a fraction. He found that (written in modern terms) the successive coefficientsc_k of(−x²)^k are to be found by multiplying the preceding coefficient by⁠m − (k − 1)/k⁠ (as in the case of integer exponents), thereby implicitly giving a formula for these coefficients. He explicitly writes the following instances^[a]

${\displaystyle (1-x^2{)}^{1/2}=1-\frac{x^2}{2}-\frac{x^4}{8}-\frac{x^6}{16}\cdots }$

${\displaystyle (1-x^2{)}^{3/2}=1-\frac{3x^2}{2}+\frac{3x^4}{8}+\frac{x^6}{16}\cdots }$

${\displaystyle (1-x^2{)}^{1/3}=1-\frac{x^2}{3}-\frac{x^4}{9}-\frac{5x^6}{81}\cdots }$

The binomial series is therefore sometimes referred to asNewton's binomial theorem. Newton gives no proof and is not explicit about the nature of the series. Later, on 1826Niels Henrik Abel discussed the subject in a paper published onCrelle's Journal, treating notably questions of convergence.^[4]

• Binomial approximation
• Binomial theorem
• Table of Newtonian series
• Lambert W function

• Abel, Niels (1826),"Recherches sur la série 1 + (m/1)x + (m(m − 1)/1.2)x² + (m(m − 1)(m − 2)/1.2.3)x³ + ...",Journal für die reine und angewandte Mathematik,1:311–339
• Coolidge, J. L. (1949), "The Story of the Binomial Theorem",The American Mathematical Monthly,56 (3):147–157,doi:10.2307/2305028,JSTOR 2305028

• Weisstein, Eric W."Binomial Series".MathWorld.
• Weisstein, Eric W."Binomial Theorem".MathWorld.
• binomial formula atPlanetMath.
• Solomentsev, E.D. (2001) [1994],"Binomial series",Encyclopedia of Mathematics,EMS Press
• "How Isaac Newton Discovered the Binomial Power Series". August 31, 2022.

• Complex analysis
• Factorial and binomial topics
• Series (mathematics)
• Real analysis

• Articles with short description
• Short description is different from Wikidata