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Mathematical identity involving sums of binomial coefficients
Abel's binomial theorem, named after Niels Henrik Abel, is a mathematical identity involving sums of binomial coefficients. It states the following: ∑
Abel's_binomial_theorem
Mathematical series
sum of the binomial series for |x| < 1. The equality extends to |x| = 1 whenever the series converges, as a consequence of Abel's theorem and by continuity
Binomial_series
Norwegian mathematician (1802–1829)
16-year-old, Abel gave a rigorous proof of the binomial theorem valid for all numbers, extending Euler's result which had held only for rationals. Abel wrote
Niels_Henrik_Abel
(graph theory) Abel's binomial theorem (combinatorics) Alspach's theorem (graph theory) Aztec diamond theorem (combinatorics) BEST theorem (graph theory)
List_of_theorems
Number of subsets of a given size
mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is
Binomial_coefficient
This is a list of factorial and binomial topics in mathematics. See also binomial (disambiguation). Abel's binomial theorem Alternating factorial Antichain
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Henrik Abel (1802–1829), a Norwegian mathematician. Abel's binomial theorem Abel elliptic functions Abel equation Abel equation of the first kind Abel–Goncharov
List of things named after Niels Henrik Abel
List_of_things_named_after_Niels_Henrik_Abel
Branch of discrete mathematics
astronomer Rabbi Abraham ibn Ezra (c. 1140) established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist
Combinatorics
Theorem in vector calculus
Stokes' theorem, also known as the Kelvin–Stokes theorem, is a theorem in vector calculus that relates the behavior of a vector field along the edge of
Stokes'_theorem
Conditions for switching order of integration in calculus
Fubini's theorem gives the conditions under which a double integral can be computed as an iterated integral, i.e. by integrating in one variable at a
Fubini's_theorem
Generalization of the product rule in calculus
The Leibniz rule bears a strong resemblance to the binomial theorem, and in fact the binomial theorem can be proven directly from the Leibniz rule by taking
General_Leibniz_rule
Septic function Octic function Completing the square Abel–Ruffini theorem Bring radical Binomial theorem Blossom (functional) Root of a function nth root
List_of_polynomial_topics
Theorem in calculus
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through
Divergence_theorem
Theorem in calculus relating line and double integrals
In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in R
Green's_theorem
Mathematical connection between field theory and group theory
the four basic arithmetic operations. This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five
Galois_theory
Theorem in mathematics
In calculus and real analysis, the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating
Mean_value_theorem
On converting relations to functions of several real variables
In multivariable calculus, the implicit function theorem is a theorem that provides sufficient conditions under which a planar curve specified by F ( x
Implicit_function_theorem
Type of polynomial sequence
the binomial theorem can be stated by saying that the sequence { x n : n = 0 , 1 , 2 , … } {\displaystyle \{x^{n}:n=0,1,2,\ldots \}} is of binomial type
Binomial_type
(Mathematical) decomposition into a product
) {\displaystyle x^{4}+x^{2}+1=(x^{2}+x+1)(x^{2}-x+1)} Binomial expansions The binomial theorem supplies patterns that can easily be recognized from the
Factorization
Statement relating differentiable symmetries to conserved quantities
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law
Noether's_theorem
Relationship between derivatives and integrals
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at every
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Theorem in mathematics
In mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that
Inverse_function_theorem
Mathematical notation
_{|\alpha |=k}{\binom {k}{\alpha }}\,x^{\alpha }} Multi-binomial theorem ( x + y ) α = ∑ ν ≤ α ( α ν ) x ν y α − ν . {\displaystyle (x+y)^{\alpha
Multi-index_notation
Method of differentiating single-term polynomials
the terms cancel. This proof only works for natural numbers as the binomial theorem only works for natural numbers. Let y = x n {\displaystyle y=x^{n}}
Power_rule
Approximation of a function by a polynomial
In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree
Taylor's_theorem
Study of rates of change
Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation and integration are inverse
Differential_calculus
Statement about integration on manifolds
generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about
Generalized_Stokes_theorem
average rate of change binomial coefficient Any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. Commonly
Glossary_of_calculus
indeterminate forms Abel's theorem – relates the limit of a power series to the sum of its coefficients Lagrange inversion theorem – gives the Taylor series
List_of_real_analysis_topics
Theorem to simplify sums of products of sequences
series Integration by parts Cesàro summation Abel's theorem Abel sum formula Chu, Wenchang (2007). "Abel's lemma on summation by parts and basic hypergeometric
Summation_by_parts
3D generalization of the Leibniz integral rule
calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or simply the Reynolds theorem, named after Osborne Reynolds
Reynolds_transport_theorem
Integral of sin(x)/x from 0 to infinity
{\displaystyle \sin t} as well as a version of Abel's theorem (a consequence of the final value theorem for the Laplace transform). Therefore, ∫ 0 ∞ sin
Dirichlet_integral
Test for series convergence
In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The test is named after
Abel's_test
Method of evaluating certain integrals along paths in the complex plane
that −1 = eπi for the second branch of the logarithm. Next we apply the binomial expansion, obtaining 1 z e 1 4 π i ( 1 − ( 1 / 4 1 ) 3 z + ( 1 / 4 2 )
Contour_integration
1941 mathematics book
problem, Theory of second-degree equations—Study of the quadratic trinomial, Binomial and trinomial equations, Sequences, Logarithms, Compound Interest, Amortization
Álgebra_de_Baldor
Popular mathematics book from 1940
problem and the binomial theorem. Chapter VIII ("Rubber-sheet Geometry") concerns concepts in topology, such as the Jordan curve theorem, the Euler characteristics
Mathematics and the Imagination
Mathematics_and_the_Imagination
Course designed to prepare students for calculus
exercised with trigonometric functions and trigonometric identities. The binomial theorem, polar coordinates, parametric equations, and the limits of sequences
Precalculus
Differential calculus on function spaces
L}{\partial x}}=0} implies that the Lagrangian is time-independent. By Noether's theorem, there is an associated conserved quantity. In this case, this quantity
Calculus_of_variations
Differentiation under the integral sign formula
integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just the particular case of the above
Leibniz_integral_rule
Mathematical theorem, used in calculus
continuous and invertible function. It follows from the intermediate value theorem that f {\displaystyle f} is strictly monotone. Consequently, f {\displaystyle
Integral_of_inverse_functions
Certain vector fields are the sum of an irrotational and a solenoidal vector field
In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector
Helmholtz_decomposition
inequality Muirhead's inequality Newton's inequalities Stein–Strömberg theorem Binomial coefficient bounds Factorial bounds XYZ inequality Fisher's inequality
List_of_inequalities
Vector operator in vector calculus
source density div v by the circulation density ∇ × v. This "decomposition theorem" is a by-product of the stationary case of electrodynamics. It is a special
Divergence
which flourished for several hundred years. "He also discovered the binomial theorem for integer exponents, which "was a major factor in the development
Timeline_of_mathematics
Algebraic structure with addition, multiplication, and division
symmetries of field extensions, provides an elegant proof of the Abel–Ruffini theorem that general quintic equations cannot be solved in radicals. Fields
Field_(mathematics)
Mathematical operation in calculus
= 0 {\displaystyle F(x,y)=0} through the point. The implicit function theorem supplies the missing justification. It asserts as follows: suppose that
Implicit_differentiation
Mathematical approximation of a function
function, which become generally more accurate as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such
Taylor_series
Multivariate derivative (mathematics)
endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous)
Gradient
Convergence test for infinite series
_{a}^{b}f'(t)\,dt=f(b)-f(a)} Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem Differential Definitions
Cauchy_condensation_test
Mathematical criterion about whether a series converges
divergence of infinite products. This can be achieved using following theorem: Let { a n } n = 1 ∞ {\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty
Convergence_tests
Test for convergence of alternating series
the monotonically decreasing sequence S2m+1, the monotone convergence theorem then implies that this sequence converges as m approaches infinity. Similarly
Alternating_series_test
Mathematical relation consisting of a multi-variable function equal to zero
Some equations do not admit an explicit solution. The implicit function theorem provides conditions under which some kinds of implicit equations define
Implicit_function
Method of mathematical integration
under the integral sign (via the monotone convergence theorem and dominated convergence theorem). While the Riemann integral considers the area under
Lebesgue_integral
Calculus of functions of several variables
is embodied by the integral theorems of vector calculus: Gradient theorem Stokes' theorem Divergence theorem Green's theorem. In a more advanced study of
Multivariable_calculus
Infinite sum
by his expansion of a complex function in such a form. Abel (1826) in his memoir on the binomial series 1 + m 1 ! x + m ( m − 1 ) 2 ! x 2 + ⋯ {\displaystyle
Series_(mathematics)
binomial theorem, which he had extended to include fractional and negative exponents. Newton succeeded in expanding the applicability of the binomial
History_of_calculus
Order in which multiple or iterated integrals are computed
iterated integrals (or multiple integrals through the use of Fubini's theorem) of functions into other, hopefully simpler, integrals by changing the
Order of integration (calculus)
Order_of_integration_(calculus)
Mathematical theorem
for the symmetry to hold are given by Schwarz's theorem, also called Clairaut's theorem or Young's theorem. In the context of partial differential equations
Symmetry of second derivatives
Symmetry_of_second_derivatives
Arithmetic operation, inverse of nth power
determine cube roots. In 1665, Isaac Newton discovered the general binomial theorem, which can convert an nth root into an infinite series. Based on approach
Nth_root
Modified summation method applicable to some divergent series
resulting limit is the same. Abel summation Abel's summation formula Abel–Plana formula Abelian and tauberian theorems Almost convergent sequence Borel
Cesàro_summation
Matrix of partial derivatives of a vector-valued function
generalization includes generalizations of the inverse function theorem and the implicit function theorem, where the non-nullity of the derivative is replaced by
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Summation method for some divergent series
z. Binomial transform Borel summation Cesàro summation Lambert summation Perron's formula Abelian and Tauberian theorems Abel–Plana formula Abel's summation
Euler_summation
Integration over a non-flat region in 3D space
and vector calculus, such as the divergence theorem, magnetic flux, and its generalization, Stokes' theorem. Let us notice that we defined the surface
Surface_integral
Mathematical method in calculus
The discrete analogue for sequences is called summation by parts. The theorem can be derived as follows. For two continuously differentiable functions
Integration_by_parts
developed by Joseph Liouville in the 1830s and 1840s, leading to Liouville's theorem which classifies which expressions have closed-form antiderivatives. A
Lists_of_integrals
Indefinite integral
Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a closed interval
Antiderivative
Calculus of functions generalization
concepts from differential geometry such as differential forms and Stokes' theorem. This extensive use of linear algebra also allows a natural generalization
Calculus_on_Euclidean_space
Point to which functions converge in analysis
advantages of working with non-deleted limits is that they allow to state the theorem about limits of compositions without any constraints on the functions (other
Limit_of_a_function
Operation on differential forms
natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus. If a differential k {\displaystyle
Exterior_derivative
Operation in calculus
this case, they are also called indefinite integrals. The fundamental theorem of calculus relates definite integration to differentiation and provides
Integral
roots) in the formula. This is due to the Abel-Ruffini theorem. On the other hand, the fundamental theorem of algebra shows that all nonconstant polynomials
Polynomial_root-finding
Formula in calculus
itself can be viewed as the polynomial remainder theorem (the little Bézout theorem, or factor theorem), generalized to an appropriate class of functions
Chain_rule
Evaluates a line integral through a gradient field using the original scalar field
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated
Gradient_theorem
Mathematical techniques used in probability theory and related fields
Clark–Ocone theorem, which allows the process in the martingale representation theorem to be identified explicitly. A simplified version of this theorem is as
Malliavin_calculus
Branch of mathematics
curves. These two branches are related to each other by the fundamental theorem of calculus. Calculus uses convergence of infinite sequences and infinite
Calculus
Branch of mathematical analysis
the product and quotient rule and has analogs to Rolle's theorem and the mean value theorem. However, this fractional derivative produces significantly
Fractional_calculus
Calculus of vector-valued functions
corresponding theorems which generalize the fundamental theorem of calculus to higher dimensions: In two dimensions, the divergence and curl theorems reduce
Vector_calculus
Integrals not expressible in closed-form from elementary functions
elementary function. A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. This theorem also provides a basis
Nonelementary_integral
Circulation density in a vector field
vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector
Curl_(mathematics)
Mathematical function with no sudden changes
{\left|f(x_{0})-y_{0}\right|}{2}}.} The intermediate value theorem is an existence theorem, based on the real number property of completeness, and states:
Continuous_function
Generalized chain rule in calculus
by collecting like terms, or alternatively, by applying the multinomial theorem. The special case f ( x ) = e x {\displaystyle f(x)=e^{x}} , g ( x ) =
Faà_di_Bruno's_formula
sums and alternating sums of binomial coefficients. It has been suggested that he may have also discovered the binomial theorem in this context. 3rd century
Timeline of scientific discoveries
Timeline_of_scientific_discoveries
American mathematician
bounded. In 1949, Agnew proved the following theorem relating the partial sums of a sequence to its Abel transform: lim sup t → 1 − | ∑ k = 1 ∞ t k u
Ralph_Palmer_Agnew
Relation between relative derivatives of three variables
cyclic relation, cyclical rule, Euler's chain rule, or the reciprocity theorem, is a formula which relates partial derivatives of three interdependent
Triple_product_rule
Type of derivative in mathematics
of the exogeneous variables, other than through the implicit function theorem, and the total derivative is handled implicitly. Thus, although "total
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Mathematical operation in calculus
_{a}^{b}f'(t)\,dt=f(b)-f(a)} Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem Differential Definitions
Logarithmic_derivative
Generalization of definite integrals to functions of multiple variables
distribution. Main analysis theorems that relate multiple integrals: Divergence theorem Stokes' theorem Green's theorem Stewart, James (2008). Calculus:
Multiple_integral
Mathematical notion of infinitesimal difference
_{a}^{b}f'(t)\,dt=f(b)-f(a)} Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem Differential Definitions
Differential_(mathematics)
Instantaneous rate of change (mathematics)
constant, because the derivative of a constant is zero. The fundamental theorem of calculus shows that finding an antiderivative of a function gives a
Derivative
Technique in integral evaluation
theorem. Alternatively, the requirement that det(Dφ) ≠ 0 can be eliminated by applying Sard's theorem. For Lebesgue measurable functions, the theorem
Integration_by_substitution
Concept in mathematical analysis
Lebesgue integral of the function. Specifically, the following theorem holds (Apostol 1974, Theorem 10.33): If a function f is Riemann integrable on [a,b] for
Improper_integral
Mathematical identities
\varphi )} in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality of mixed partials). This result is a special
Vector_calculus_identities
Change of variable for integrals involving trigonometric functions
_{a}^{b}f'(t)\,dt=f(b)-f(a)} Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem Differential Definitions
Tangent half-angle substitution
Tangent_half-angle_substitution
Notation of differential calculus
Cauchy condensation Dirichlet Abel Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence
Notation_for_differentiation
Method for constructing existence proofs and calculating solutions in variational calculus
space W {\displaystyle W} . In this case the sequential Banach–Alaoglu theorem implies that any bounded sequence ( u n ) {\displaystyle (u_{n})} in V
Direct method in the calculus of variations
Direct_method_in_the_calculus_of_variations
Method for partial-fraction expansion
has fractional expressions where some factors may repeat as powers of a binomial. In integral calculus we would want to write a fractional algebraic expression
Heaviside_cover-up_method
Vector calculus formulas relating the bulk with the boundary of a region
mathematician George Green, who discovered Green's theorem. This identity is derived from the divergence theorem applied to the vector field F = ψ ∇φ while using
Green's_identities
Differential operator in mathematics
where n is the outward unit normal to the boundary of V. By the divergence theorem, ∫ V div ∇ u d V = ∫ S ∇ u ⋅ n d S = 0. {\displaystyle \int _{V}\operatorname
Laplace_operator
Instantaneous rate of change of the function
_{a}^{b}f'(t)\,dt=f(b)-f(a)} Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem Differential Definitions
Directional_derivative
Derivative defined on normed spaces
Let D {\displaystyle D} be any linear functional. Riesz Representation Theorem tells us that D {\displaystyle D} could be defined by D v = ⟨ a , v ⟩ {\displaystyle
Fréchet_derivative
ABELS BINOMIAL-THEOREM
ABELS BINOMIAL-THEOREM
ABELS BINOMIAL-THEOREM
ABELS BINOMIAL-THEOREM
ABELS BINOMIAL-THEOREM
ABELS BINOMIAL-THEOREM
ABELS BINOMIAL-THEOREM
ABELS BINOMIAL-THEOREM
ABELS BINOMIAL-THEOREM