Binomial Series
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the binomial series is a generalization of the polynomial that comes from a
binomial formula In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
expression like (1+x)^n for a nonnegative integer n. Specifically, the binomial series is the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
for the
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
f(x)=(1+x)^ centered at x = 0, where \alpha \in \Complex and , x, < 1. Explicitly, where the
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
on the right-hand side of () is expressed in terms of the (generalized) binomial coefficients :\binom := \frac.


Special cases

If is a nonnegative
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, then the th term and all later terms in the series are 0, since each contains a factor ; thus in this case the series is finite and gives the algebraic
binomial formula In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
. Closely related is the ''negative binomial series'' defined by the Taylor series for the function g(x)=(1-x)^ centered at x = 0, where \alpha \in \Complex and , x, < 1. Explicitly, :\begin \frac &= \sum_^ \; \frac \; x^k \\ &= 1 + \alpha x + \frac x^2 + \frac x^3 + \cdots, \end which is written in terms of the multiset coefficient :\left(\!\!\!\!\right) := = \frac\,.


Convergence


Conditions for convergence

Whether () converges depends on the values of the complex numbers and . More precisely: #If , the series converges absolutely for any complex number . #If , the series converges absolutely
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
either or , where denotes the
real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of . # If and , the series converges if and only if . #If , the series converges if and only if either or . #If , the series diverges, unless is a non-negative integer (in which case the series is a finite sum). In particular, if \alpha 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.


Identities to be used in the proof

The following hold for any complex number : : = 1, Unless \alpha is a nonnegative integer (in which case the binomial coefficients vanish as k is larger than \alpha), a useful asymptotic relationship for the binomial coefficients is, in
Landau notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
: This is essentially equivalent to Euler's definition of the
Gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
: :\Gamma(z) = \lim_ \frac, and implies immediately the coarser bounds for some positive constants and . Formula () for the generalized binomial coefficient can be rewritten as


Proof

To prove (i) and (v), apply the
ratio test In mathematics, the ratio test is a test (or "criterion") for the convergence of a series :\sum_^\infty a_n, where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert a ...
and use formula () above to show that whenever \alpha is not a nonnegative integer, the
radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
is exactly 1. Part (ii) follows from formula (), by comparison with the -series : \sum_^\infty \; \frac , with p=1+\operatorname\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 \operatorname \alpha> - 1 is assumed. On the other hand, the series does not converge if , x, =1 and \operatorname \alpha \le - 1 , again by formula (). Alternatively, we may observe that for all j, \left , \frac - 1 \right , \ge 1 - \frac \ge 1 . Thus, by formula (), for all k, \left, \ \ge 1 . This completes the proof of (iii). Turning to (iv), we use identity () above with x=-1 and \alpha-1 in place of \alpha, along with formula (), to obtain :\sum_^n \; \; (-1)^k = \;(-1)^n= \frac (1+o(1)) as n\to\infty. Assertion (iv) now follows from the asymptotic behavior of the sequence n^ = e^. (Precisely, \left, e^\ = e^ certainly converges to 0 if \operatorname\alpha>0 and diverges to +\infty if \operatorname\alpha<0. If \operatorname\alpha=0, then n^ = e^ converges if and only if the sequence \operatorname\alpha\log n converges \bmod, which is certainly true if \alpha=0 but false if \operatorname\alpha \neq0: in the latter case the sequence is dense \bmod, due to the fact that \log n diverges and \log (n+1)-\log n 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 and using formula (), one has that the sum of the series is an
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
solving the
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
with initial data . The unique solution of this problem is the function , which is therefore the sum of the binomial series, at least for . The equality extends to whenever the series converges, as a consequence of
Abel's theorem In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel. Theorem Let the Taylor series G (x) = \sum_^\infty a_k x^k be a pow ...
and by continuity of .


History

The first results concerning binomial series for other than positive-integer exponents were given by Sir
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
in the study of
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...
s 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-\frac2-\frac8-\frac\cdots :(1-x^2)^=1-\frac2+\frac8+\frac\cdots :(1-x^2)^=1-\frac3-\frac9-\frac\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 Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
discussed the subject in a paper published on '' Crelle's Journal'', treating notably questions of convergence.


See also

*
Binomial approximation The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number ''x''. It states that : (1 + x)^\alpha \approx 1 + \alpha x. It is valid when , x, -1 and \alpha \geq 1. Derivations Using linear ...
*
Binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
*
Table of Newtonian series In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence a_n written in the form :f(s) = \sum_^\infty (-1)^n a_n = \sum_^\infty \frac a_n where : is the binomial coefficient and (s)_n is the falling factorial. N ...


Footnotes


Notes


Citations


References

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External links

* * * * * {{Calculus topics Complex analysis Factorial and binomial topics Mathematical series Real analysis