Английская Википедия:Binomial 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 <math>f(x)=(1+x)^{\alpha}</math>, where <math>\alpha \in \Complex</math> and <math>|x| < 1</math>. Explicitly,
where the power series on the right-hand side of (Шаблон:EquationNote) is expressed in terms of the (generalized) binomial coefficients
- <math>\binom{\alpha}{k} := \frac{\alpha (\alpha-1) (\alpha-2) \cdots (\alpha-k+1)}{k!}. </math>
Note that if Шаблон:Mvar is a nonnegative integer Шаблон:Mvar then the Шаблон:Math term and all later terms in the series are Шаблон:Math, since each contains a factor of Шаблон:Math. Thus, in this case, the series is finite and gives the algebraic binomial formula.
Convergence
Conditions for convergence
Whether (Шаблон:EquationNote) converges depends on the values of the complex numbers Шаблон:Mvar and Шаблон:Mvar. More precisely:
- If Шаблон:Math, the series converges absolutely for any complex number Шаблон:Mvar.
- If Шаблон:Math, the series converges absolutely if and only if either Шаблон:Math or Шаблон:Math, where Шаблон:Math denotes the real part of Шаблон:Mvar.
- If Шаблон:Math and Шаблон:Math, the series converges if and only if Шаблон:Math.
- If Шаблон:Math, the series converges if and only if either Шаблон:Math or Шаблон:Math.
- If Шаблон:Math, the series diverges except when Шаблон:Mvar is a non-negative integer, in which case the series is a finite sum.
In particular, if Шаблон:Mvar is not a non-negative integer, the situation at the boundary of the disk of convergence, Шаблон:Math, is summarized as follows:
- If Шаблон:Math, the series converges absolutely.
- If Шаблон:Math, the series converges conditionally if Шаблон:Math and diverges if Шаблон:Math.
- If Шаблон:Math, the series diverges.
Identities to be used in the proof
The following hold for any complex number Шаблон:Mvar:
- <math>{\alpha \choose 0} = 1,</math>
Шаблон:NumBlk 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 relationship for the binomial coefficients is, in Landau notation:
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
Шаблон:NumBlk\le \left|{\alpha \choose k}\right| \le \frac {M} {k^{1+\operatorname{Re}\alpha}}, </math>|Шаблон:EquationRef}} for some positive constants Шаблон:Mvar and Шаблон:Mvar .
Formula (Шаблон:EquationNote) for the generalized binomial coefficient can be rewritten as Шаблон:NumBlk
Proof
To prove (i) and (v), apply the ratio test and use formula (Шаблон:EquationNote) above to show that whenever <math>\alpha</math> is not a nonnegative integer, the radius of convergence is exactly 1. Part (ii) follows from formula (Шаблон:EquationNote), by comparison with the [[Convergence tests#p-series test|Шаблон:Mvar-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 (Шаблон:EquationNote) to obtain
and then use (ii) and formula (Шаблон:EquationNote) 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 (Шаблон:EquationNote). 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 (Шаблон:EquationNote), 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 (Шаблон:EquationNote) above with <math>x=-1</math> and <math>\alpha-1</math> in place of <math>\alpha</math>, along with formula (Шаблон:EquationNote), 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. Differentiating term-wise the binomial series within the disk of convergence Шаблон:Math and using formula (Шаблон:EquationNote), one has that the sum of the series is an analytic function solving the ordinary differential equation Шаблон:Math with initial condition Шаблон:Math.
The unique solution of this problem is the function Шаблон:Math. Indeed, multiplying by the integrating factor Шаблон:Math 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 Шаблон:Math is a constant, which the initial condition tells us is Шаблон:Math. That is, Шаблон:Math is the sum of the binomial series for Шаблон:Math.
The equality extends to Шаблон:Math whenever the series converges, as a consequence of Abel's theorem and by continuity of Шаблон:Math.
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 Шаблон:Mvar is a positive integer, several common sequences are apparent. The case Шаблон:Math gives the series Шаблон:Math, where the coefficient of each term of the series is simply Шаблон:Math. The case Шаблон:Math gives the series Шаблон:Math, which has the counting numbers as coefficients. The case Шаблон:Math gives the series Шаблон:Math, which has the triangle numbers as coefficients. The case Шаблон:Math gives the series Шаблон:Math, which has the tetrahedral numbers as coefficients, and similarly for higher integer values of Шаблон:Mvar.
The negative binomial series includes the case of the geometric series, the power series[1] <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.[2]
History
The first results concerning binomial series for other than positive-integer exponents were given by Sir Isaac Newton in the study of areas enclosed under certain curves. John Wallis built upon this work by considering expressions of the form Шаблон:Math where Шаблон:Mvar is a fraction. He found that (written in modern terms) the successive coefficients Шаблон:Math of Шаблон:Math are to be found by multiplying the preceding coefficient by Шаблон:Sfrac (as in the case of integer exponents), thereby implicitly giving a formula for these coefficients. He explicitly writes the following instancesШаблон:Efn
- <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 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. Шаблон:Sfn
See also
Footnotes
Notes
Citations
References
External links
- ↑ Шаблон:Citation
- ↑ Шаблон:Citation, §22.
