Binomial series

 ( Complex Analysis ) 

In mathematics, the binomial series is a generalization of the binomial formula to cases where the exponent is not a positive integer:

where $\backslash alpha$ is any complex number, and the power series on the right-hand side is expressed in terms of the (generalized) binomial coefficients

$\backslash binom\{\backslash alpha\}\{k\}\; =\; \backslash frac\{\backslash alpha\; (\backslash alpha-1)\; (\backslash alpha-2)\; \backslash cdots\; (\backslash alpha-k+1)\}\{k!\}.$

The binomial series is the MacLaurin series for the function $f(x)=(1+x)^\backslash alpha$. It converges when .

If is a nonnegative integer then the term and all later terms in the series are , since each contains a factor of . In this case, the series is a finite polynomial, equivalent to the binomial formula.

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Convergence

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Conditions for convergence Whether () converges depends on the values of the complex numbers and . More precisely:

1. If , the series converges absolutely for any complex number .
2. If , the series converges absolutely if and only if either or , where denotes the complex number of .
3. If and , the series converges if and only if .
4. If , the series converges if and only if either or .
5. If , the series divergent series except when is a non-negative integer, in which case the series is a finite sum.

In particular, if is not a non-negative integer, the situation at the boundary of the disk of convergence, , is summarized as follows:

• If , the series converges absolutely.
• If , the series converges conditionally if and diverges if .
• If , the series diverges.

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Identities to be used in the proof The following hold for any complex number :

$\{\backslash alpha\; \backslash choose\; 0\}\; \backslash !=\; 1,$

Unless $\backslash alpha$ is a nonnegative integer (in which case the binomial coefficients vanish as $k$ is larger than $\backslash alpha$), a useful asymptotic relationship for the binomial coefficients is, in Landau notation:

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

$\backslash Gamma(z)\; =\; \backslash lim\_\{k\; \backslash to\; \backslash infty\}\; \backslash frac\{k!\; \backslash ,k^z\}\{z(z+1)\backslash cdots(z+k)\},$

and implies immediately the coarser bounds

for some positive constants and .

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

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Proof To prove (i) and (v), apply the ratio test and use formula () above to show that whenever $\backslash alpha$ is not a nonnegative integer, the radius of convergence is exactly 1. Part (ii) follows from formula (), by comparison with the -series

$\backslash sum\_\{k=1\}^\backslash infty\; \backslash frac1\{k^p\},$

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

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

$\backslash sum\_\{k=0\}^n\; \backslash !\{\backslash alpha\backslash choose\; k\}\; (-1)^k\; =\; \backslash !\{\backslash alpha-1\; \backslash choose\; n\}\; (-1)^n=\; \backslash frac1\{\backslash Gamma(-\backslash alpha+1)n^\backslash alpha\}\; (1+o(1))$

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

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Summation of the binomial series The usual argument to compute the sum of the binomial series goes as follows. Derivative term-wise the binomial series within the disk of convergence and using formula (), one has that the sum of the series is an analytic function solving the ordinary differential equation with initial condition .

The unique solution of this problem is the function . Indeed, multiplying by the integrating factor gives

$0=(1+x)^\{-\backslash alpha\}u\text{'}(x)\; -\; \backslash alpha\; (1+x)^\{-\backslash alpha-1\}\; u(x)=\; \backslash big(1+x)^\{-\backslash alpha\}u(x)\backslash big\text{'}\backslash ,,$

so the function is a constant, which the initial condition tells us is . That is, is the sum of the binomial series for .

The equality extends to whenever the series converges, as a consequence of Abel's theorem and by continuity of .

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Negative binomial series Closely related is the negative binomial series defined by the MacLaurin series for the function $g(x)=(1-x)^\{-\backslash alpha\}$, where $\backslash alpha\; \backslash in\; \backslash Complex$ and . Explicitly,

$\backslash begin\{align\}$

\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}
     

which is written in terms of the multiset coefficient

$\backslash left(\backslash !\backslash !\{\backslash alpha\backslash choose\; k\}\backslash !\backslash !\backslash right)\; =\; \{\backslash alpha+k-1\; \backslash choose\; k\}\; =\; \backslash frac\{\backslash alpha\; (\backslash alpha+1)\; (\backslash alpha+2)\; \backslash cdots\; (\backslash alpha+k-1)\}\{k!\}\backslash ,.$

When is a positive integer, several common sequences are apparent. The case gives the series , where the coefficient of each term of the series is simply . The case gives the series , which has the counting numbers as coefficients. The case gives the series , which has the triangle numbers as coefficients. The case gives the series , which has the tetrahedral numbers as coefficients, and similarly for higher integer values of .

The negative binomial series includes the case of the geometric series, the power series $$\backslash frac\{1\}\{1-x\}\; =\; \backslash sum\_\{n=0\}^\backslash infty\; x^n$$ (which is the negative binomial series when $\backslash alpha=1$, convergent in the disc ) and, more generally, series obtained by differentiation of the geometric power series: $$\backslash frac\{1\}\{(1-x)^n\}\; =\; \backslash frac\{1\}\{(n-1)!\}\backslash frac\{d^\{n-1\}\}\{dx^\{n-1\}\}\backslash frac\{1\}\{1-x\}$$ with $\backslash alpha=n$, a positive integer., §22.

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History The first results concerning binomial series for other than positive-integer exponents were given by Sir Isaac Newton in the study of enclosed under certain curves. John Wallis built upon this work by considering expressions of the form where is a fraction. He found that (written in modern terms) the successive coefficients of are to be found by multiplying the preceding coefficient by (as in the case of integer exponents), thereby implicitly giving a formula for these coefficients. He explicitly writes the following instances

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

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

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

The binomial series is therefore sometimes referred to as 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.

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See also

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

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Footnotes

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Notes

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Citations

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External links

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