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
for a nonnegative integer
. 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 ...
centered at
, where
and
. 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
:
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
centered at
, where
and
. Explicitly,
:
which is written in terms of the
multiset coefficient
:
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
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 :
:
Unless
is a nonnegative integer (in which case the binomial coefficients vanish as
is larger than
), 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 ...
:
:
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
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
:
with
. To prove (iii), first use formula () to obtain
and then use (ii) and formula () again to prove convergence of the right-hand side when
is assumed. On the other hand, the series does not converge if
and
, again by formula (). Alternatively, we may observe that for all
,
. Thus, by formula (), for all
. This completes the proof of (iii). Turning to (iv), we use identity () above with
and
in place of
, along with formula (), to obtain
:
as
. Assertion (iv) now follows from the asymptotic behavior of the sequence
. (Precisely,
certainly converges to
if
and diverges to
if
. If
, then
converges if and only if the sequence
converges
, which is certainly true if
but false if
: in the latter case the sequence is dense
, due to the fact that
diverges and
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
:
:
:
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
*
*
External links
*
*
*
*
*
{{Calculus topics
Complex analysis
Factorial and binomial topics
Mathematical series
Real analysis