Binomial series
Encyclopedia
In mathematics
, the binomial series is the Taylor series
at x = 0 of the function f given by f(x) = (1 + x) α, where is an arbitrary complex number
. Explicitly,
and the binomial series is the power series on the right hand side of (1), expressed in terms of the (generalized) binomial coefficient
s
.
The following variant holds for arbitrary complex β, but is especially useful for handling negative integer exponents in (1):
To prove it, substitute x = −z in (1) and apply a binomial coefficient identity.
Unless α is a nonnegative integer (in which case the binomial coefficients become 0 as k → ∞), a useful asymptotic
relationship for the binomial coefficients is, in Landau notation:
This is essentially equivalent to Euler's definition of the Gamma function
:
and implies immediately the coarser asymptotics
as k → ∞, which is sufficient for our needs. The simple bound (4) may also be obtained by means of elementary inequalities (see the addendum below).
is exactly 1. The absolute convergence
(ii) follows from formula (4), by comparison with the p-series
with p = 1 + Re(α). To prove (iii), first use formula (3) to obtain
and then use (ii) and formula (4) again to prove convergence of the right-hand side when Re(α) > −1 is assumed. On the other hand, the series does not converge if |x| = 1 and Re(α) ≤ −1, because in that case, for all k,
solving the ordinary differential equation (1 + x)u' (x) = α u(x) with initial data u(0) = 1. The unique solution of this problem is the function u(x) = (1 + x)α, which is therefore the sum of the binomial series, at least for |x| < 1. The equality extends to |x| = 1 whenever the series converges, as a consequence of Abel's theorem
and by continuity of (1 + x)α.
in the study of areas enclosed under certain curves. Extending work by John Wallis who calculated such areas for y = (1 − x2)n with n = 0, 1, 2, 3, ... he considered fractional exponents. He found for such exponent m that (in modern formulation) the successive coefficients ck of (−x2)k 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; most likely he verified instances treating the series as (again in modern terminology) formal power series
. Later, Niels Henrik Abel
treated the subject in a memoir, treating notably questions of convergence.
with
as follows. By the inequality of arithmetic and geometric means
Using the expansion
the latter arithmetic mean writes
To estimate its kth power we then use the inequality
that holds true for any real number r as soon as 1 + r/k ≥ 0. Moreover, we have elementary bounds for the sums:
Thus,
with
proving the claim.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the binomial series is the Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
at x = 0 of the function f given by f(x) = (1 + x) α, where is an arbitrary complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
. Explicitly,
and the binomial series is the power series on the right hand side of (1), expressed in terms of the (generalized) binomial coefficient
Binomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
s
Special cases
If α is a nonnegative integer n, then the (n + 1)th term and all later terms in the series are 0, since each contains a factor (n − n); thus in this case the series is finite and gives the algebraic binomial formulaBinomial theorem
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...
.
The following variant holds for arbitrary complex β, but is especially useful for handling negative integer exponents in (1):
To prove it, substitute x = −z in (1) and apply a binomial coefficient identity.
Conditions for convergence
Whether (1) converges depends on the values of the complex numbers and . More precisely:- If , the series converges for any complex number α.
- If , the series converges absolutely if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
either or .
- If and , the series converges if and only if .
- If , the series diverges, unless is a non-negative integer (in which case the series is finite).
Identities to be used in the proof
The following hold for any complex number α:Unless α is a nonnegative integer (in which case the binomial coefficients become 0 as k → ∞), a useful asymptotic
Asymptotic analysis
In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
relationship for the binomial coefficients is, in Landau notation:
This is essentially equivalent to Euler's definition of the Gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
:
and implies immediately the coarser asymptotics
as k → ∞, which is sufficient for our needs. The simple bound (4) may also be obtained by means of elementary inequalities (see the addendum below).
Sketch of proof
To prove (i), apply the ratio test and use formula (2) above to show that whenever α is not a nonnegative integer, the radius of convergenceRadius of convergence
In mathematics, the radius of convergence of a power series is a quantity, either a non-negative real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well...
is exactly 1. The absolute convergence
Absolute convergence
In mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...
(ii) follows from formula (4), by comparison with the p-series
with p = 1 + Re(α). To prove (iii), first use formula (3) to obtain
and then use (ii) and formula (4) again to prove convergence of the right-hand side when Re(α) > −1 is assumed. On the other hand, the series does not converge if |x| = 1 and Re(α) ≤ −1, because in that case, for all k,
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 convergence disk |x| < 1 and using formula (1), one has that the sum of the series is an analytic functionAnalytic 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, categories that are similar in some ways, but different in others...
solving the ordinary differential equation (1 + x)u
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:...
and by continuity of (1 + x)α.
History
The first results concerning binomial series for other than positive-integer exponents were given by Sir Isaac NewtonIsaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
in the study of areas enclosed under certain curves. Extending work by John Wallis who calculated such areas for y = (1 − x2)n with n = 0, 1, 2, 3, ... he considered fractional exponents. He found for such exponent m that (in modern formulation) the successive coefficients ck of (−x2)k 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 instancesThe 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; most likely he verified instances treating the series as (again in modern terminology) formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
. Later, Niels Henrik Abel
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
treated the subject in a memoir, treating notably questions of convergence.
Elementary bounds on the coefficients
In order to keep the whole discussion within elementary methods, one may derive the asymptotics (3) proving the inequalitywith
as follows. By the inequality of arithmetic and geometric means
Using the expansion
the latter arithmetic mean writes
To estimate its kth power we then use the inequality
that holds true for any real number r as soon as 1 + r/k ≥ 0. Moreover, we have elementary bounds for the sums:
Thus,
with
proving the claim.
See also
- Binomial theorem
- Table of Newtonian series