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Summation method for some divergent series
In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series
Euler_summation
Summation formula
Cesàro summation Euler summation Gauss–Kronrod quadrature formula Darboux's formula Euler–Boole summation List of topics named after Leonhard Euler Apostol
Euler–Maclaurin_formula
Modified summation method applicable to some divergent series
Divergent series Euler summation Euler–Boole summation Fejér's theorem Hölder summation Lambert summation Perron's formula Ramanujan summation Riesz mean Silverman–Toeplitz
Cesàro_summation
fraction Euler product formula for the Riemann zeta function. Euler–Maclaurin formula (Euler's summation formula) relating integrals to sums Euler–Rodrigues
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Summation method for some divergent series
In mathematics, Euler–Boole summation is a method for summing alternating series. The concept is named after Leonhard Euler and George Boole. Boole published
Euler–Boole_summation
Addition of several numbers or other values
Euler–Maclaurin formula. For summations in which the summand is given (or can be interpolated) by an integrable function of the index, the summation can
Summation
Infinite series that is not convergent
} Ramanujan summation is a method of assigning a value to divergent series used by Ramanujan and based on the Euler–Maclaurin summation formula. The
Divergent_series
Infinite series with alternating signs
interpretations of Euler's attempts. Many of these summability methods easily assign to 1 − 2 + 3 − 4 + ... a "value" of 1/4. Cesàro summation is one of the
1_−_2_+_3_−_4_+_⋯
Infinite series that diverges
associated with the value 1/3, which is the result of applying various summation methods to the series, as well as the limit of the series using the 2-adic
1_−_2_+_4_−_8_+_⋯
Mathematical techniques for summing divergent infinite series
summation functions as a property of partial sums. If we take the Euler–Maclaurin summation formula together with the correction rule using Bernoulli numbers
Ramanujan_summation
Swiss mathematician (1707–1783)
Chapter 14: Euler's derivation of the Euler–Maclaurin summation formula. Mills, Stella (1985). "The independent derivations by Leonhard Euler and Colin
Leonhard_Euler
Summation method for divergent series
special case of Mittag-Leffler summation with α = 1. (wB) can be seen as the limiting case of generalized Euler summation method (E,q) in the sense that
Borel_summation
Analytic function in mathematics
conjectured by Konrad Knopp in 1926 and proven by Helmut Hasse in 1930 (cf. Euler summation): ζ ( s ) = 1 1 − 2 1 − s ∑ n = 0 ∞ 1 2 n + 1 ∑ k = 0 n ( n k ) ( −
Riemann_zeta_function
Second-order partial differential equation describing motion of mechanical system
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose
Euler–Lagrange_equation
Approach to finding numerical solutions of ordinary differential equations
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary
Euler_method
Concept in differential geometry
{\displaystyle g_{2}} . (The appearance of χ {\displaystyle \chi } in the summation is the usual Euler characteristic.) If the action is free, the sum has only a single
Euler characteristic of an orbifold
Euler_characteristic_of_an_orbifold
Extension of the factorial function
}t^{z-1}e^{-t}\,dt} converges absolutely, and is known as the Euler integral of the second kind. (Euler's integral of the first kind is the beta function.) The
Gamma_function
Difference between logarithm and harmonic series
577 … {\displaystyle \gamma _{0}=\gamma =0.577\dots } Euler–Lehmer constants are given by summation of inverses of numbers in a common modulo class: γ (
Euler's_constant
Special constant related to the exponential integral
In mathematics, the Gompertz constant or Euler–Gompertz constant, denoted by δ {\displaystyle \delta } , appears in integral evaluations and as a value
Gompertz_constant
Transformation of a mathematical sequence
become much smaller, much more rapidly, thus allowing rapid numerical summation. The Euler transform can be generalized (Borisov B. and Shkodrov V., 2007):
Binomial_transform
Rational number sequence
Bernoulli umbra Bell number Euler number Genocchi number Kummer's congruences Poly-Bernoulli number Hurwitz zeta function Euler summation Stirling polynomial
Bernoulli_number
Inverse of a finite difference
sum. Laplace's summation formula, closely related to the Gregory summation formula, can be seen as the discrete counterpart to the Euler–Maclaurin formula
Indefinite_sum
Shorthand notation for tensor operations
(also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms
Einstein_notation
Expression which is not assigned an interpretation
internally consistent and practically useful. For example, Ramanujan summation may seem unintuitive, as it works upon divergent series that assign finite
Undefined_(mathematics)
Formula to quantify column buckling under a given load
Euler's critical load or Euler's buckling load is the compressive load at which a slender column will suddenly bend or buckle. It is given by the formula:
Euler's_critical_load
Abel summation Cesàro summation Lindelöf summation Euler summation Borel summation Mittag-Leffler summation Lambert summation Euler–Boole summation and
Antilimit
Divergent series
with its reliance on complex analysis, and Ramanujan summation, with its shortcut to the Euler–Maclaurin formula. Instead, the method operates directly
1_+_2_+_3_+_4_+_⋯
Infinite series summing alternating 1 and -1 terms
summed by the more general fractional (C, a) methods. Abel summation is similar to Euler's attempted definition of sums of divergent series, but it avoids
Grandi's_series
Theorem bounding the growth rate of analytic functions
} Divergent series Borel summation Euler summation Cesàro summation Lambert summation Mittag-Leffler summation Phragmén–Lindelöf principle Abelian
Nachbin's_theorem
Sum of inverse squares of natural numbers
hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem. The Basel problem asks for the precise summation of the reciprocals
Basel_problem
Divergent series that can be summed by Borel summation
assigned a value of approximately 0.596347 by Borel summation. This series was first considered by Euler, who applied summability methods to assign a finite
1_−_1_+_2_−_6_+_24_−_120_+_⋯
Negative integer two units from the origin in mathematics
−341... Though divergent, Euler assigned the value 1 3 {\displaystyle {\frac {1}{3}}} to this series, known as Euler summation. The negative second power
−2
Polynomial sequence
"Worpitzky's Identity". MathWorld. Weisstein, Eric W. "Second-Order Eulerian Triangle". MathWorld. Euler-matrix (generalized rowindexes, divergent summation)
Eulerian_number
Generalization of Euler equations
fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of general
Relativistic_Euler_equations
Infinite series whose terms alternate in sign
series acceleration techniques. One of the oldest techniques is that of Euler summation, and there are many modern techniques that can offer even more rapid
Alternating_series
Operator encoding information about iterated map
method Gaspard, Pierre (1992). "r-adic one dimensional maps and the Euler summation formula". J. Phys. A: Math. Gen. 25 (8): L483–L485. Bibcode:1992JPhA
Transfer_operator
Function defined by a hypergeometric series
Arithmetica Infinitorum. Hypergeometric series were studied by Leonhard Euler, but the first full systematic treatment was given by Carl Friedrich Gauss (1813)
Hypergeometric_function
Infinite series that diverges
example, many summation methods are used in mathematics to assign numerical values even to divergent series. In particular, the Ramanujan summation of this
1_+_2_+_4_+_8_+_⋯
Numeral system in which every non-negative integer can be represented in exactly one way
d_{k-1}d_{k-1}d_{k-1}={\overline {d_{k-1}}}} . This is because the Euler summation g ( d k − 1 ¯ ) = ∑ i = 0 ∞ f ( d k − 1 ) k i = − k − 1 k − 1 = − 1
Bijective_numeration
Signed odd unit fractions sum to π/4
a finite decimal fraction. The formula is a special case of the Euler–Boole summation formula for alternating series, providing yet another example of
Leibniz_formula_for_π
Cesàro summation Euler summation Lambert summation Borel summation Summation by parts – transforms the summation of products of into other summations Cesàro
List_of_real_analysis_topics
Infinite sum
finance. Among the Ancient Greeks, the idea that a potentially infinite summation could produce a finite result was considered paradoxical, most famously
Series_(mathematics)
Summability method for a class of divergent series
In mathematical analysis and analytic number theory, Lambert summation is a summability method for summing infinite series related to Lambert series specially
Lambert_summation
Divergent sum of positive unit fractions
indices. Although the harmonic series is divergent, its Ramanujan summation has the Euler–Mascheroni constant γ {\displaystyle \gamma } as its finite
Harmonic_series_(mathematics)
Doubling map on the unit interval
ISBN 0-7923-5564-4 Pierre Gaspard, "r-adic one-dimensional maps and the Euler summation formula", Journal of Physics A, 25 (letter) L483-L485 (1992). M. Bosschaert;
Dyadic_transformation
Conjecture on zeros of the zeta function
{1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots } Leonhard Euler considered this series in the 1730s for real values of s {\displaystyle
Riemann_hypothesis
Number, approximately 3.14
"Estimating π" (PDF). How Euler Did It. Reprinted in How Euler Did Even More. Mathematical Association of America. 2014. pp. 109–118. Euler, Leonhard (1755).
Pi
Series of functions in mathematics
negative powers. Asymptotic series commonly occur when using the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin
Asymptotic_expansion
Random process of binary (boolean) random variables
ISBN 978-1-84800-047-6. Pierre Gaspard, "r-adic one-dimensional maps and the Euler summation formula", Journal of Physics A, 25 (letter) L483-L485 (1992). Dean
Bernoulli_process
Mathematical symbol representing infinity
a potential infinity. For instance, in mathematical expressions with summations and limits such as ∑ n = 0 ∞ 1 2 n = lim x → ∞ 2 x − 1 2 x − 1 = 2 , {\displaystyle
Infinity_symbol
Theorem in number theory
In mathematics, Euler's pentagonal number theorem relates the product and series representations of the Euler function. It states that ∏ n = 1 ∞ ( 1 −
Pentagonal_number_theorem
Mathematisch Centrum, (Amsterdam, 1965) pp. 51-60 Values calculated via the J expression 'b11.8'8!:2-:&(}:+}.)^:n+/\(_1^n)*%1+2*n=.i.13 Euler summation
Van Wijngaarden transformation
Van_Wijngaarden_transformation
Mechanical oscillations about an equilibrium point
even a complex structure such as an automobile body can be modeled as a "summation" of simple mass–spring–damper models. The mass–spring–damper model is
Vibration
Summation formula
infinite series. It is a generalization to the complex plane of the Euler–Maclaurin summation formula, which is used for similar purposes and derived in a similar
Darboux's_formula
Summation formula in Mathematics
indentation on the left and right of 0. Euler–Maclaurin summation formula Euler–Boole summation Ramanujan summation Hermite, C. (1901). "Extrait de quelques
Abel–Plana_formula
Millennium Prize Problem
Tristan Buckmaster, who had derived a set of closely related results on the Euler equations used in the work. The method used to generate the solution to
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
introduced by Jacob Bernoulli in 1683. More than half a century later, Euler, who had been a student of Jacob's younger brother Johann, proved that e
Proof_that_e_is_irrational
Branch of mathematics
representation of s P ( t ) {\displaystyle s_{_{P}}(t)} in terms of a summation of a potentially infinite number of harmonically related sinusoids or
Fourier_analysis
Property of a mass in motion
system is p = m v cm . {\displaystyle p=mv_{\text{cm}}.} This is known as Euler's first law. If the net force F applied to a particle is constant, and is
Momentum
Representation of mechanical stress at every point within a deformed 3D object
principle of conservation of angular momentum, equilibrium requires that the summation of moments with respect to an arbitrary point is zero, which leads to
Cauchy_stress_tensor
Integral transform useful in probability theory, physics, and engineering
law of the Jacobi theta function, which is readily proved via Poisson summation, to the functional equation. Hjalmar Mellin was among the first to study
Laplace_transform
System of symbolic representation
Descartes, Isaac Newton, Gottfried Wilhelm Leibniz, and overall Leonhard Euler. The use of many symbols is the basis of mathematical notation. They play
Mathematical_notation
Matrix representing a Euclidean rotation
{\displaystyle y=r\sin \phi } , then the above equations become the trigonometric summation angle formulae: R v = r [ cos ϕ cos θ − sin ϕ sin θ cos ϕ sin
Rotation_matrix
Mathematical notation
bracket allows using capital-sigma notation without restriction on the summation index. That is, for any property P ( k ) {\displaystyle P(k)} of the integer
Iverson_bracket
Relation between pairs of arithmetic functions
arithmetic functions by repeatedly applying the first summation. For example, if one starts with Euler's totient function φ, and repeatedly applies the transformation
Möbius_inversion_formula
Energy held by an object because of its position relative to other objects
Gravitational potential summation U = − m ( G M 1 r 1 + G M 2 r 2 ) {\displaystyle U=-m\left(G{\frac {M_{1}}{r_{1}}}+G{\frac {M_{2}}{r_{2}}}\right)}
Potential_energy
Function whose domain is the positive integers
examples. Summation functions "smooth out" these fluctuations. In some cases it may be possible to find asymptotic behaviour for the summation function
Arithmetic_function
Degen's eight-square identity Difference of two squares Euler's four-square identity Euler's identity Fibonacci's identity see Brahmagupta–Fibonacci identity
List of mathematical identities
List_of_mathematical_identities
manifolds into a single new one. It is a symplectic version of connected summation along a submanifold, often called a fiber sum. The symplectic sum is the
Symplectic_sum
Textbook by Ronald Graham, Donald Knuth, and Oren Patashnik
Concrete Mathematics as a test case for the AMS Euler typeface and Concrete Roman font. Recurrent Problems Summation Integer Functions Number Theory Binomial
Concrete_Mathematics
Euler x′ prime symbol (for derivative) 1748 Leonhard Euler Σ summation symbol 1755 Leonhard Euler ∝ proportionality sign 1768 William Emerson ∂ partial
Table of mathematical symbols by introduction date
Table_of_mathematical_symbols_by_introduction_date
Mathematical formula in harmonic analysis
either side. It can be regarded as a Poisson summation formula for non-abelian groups. The Voronoi (summation) formula for GL(2) has long been a standard
Voronoi_formula
Mathematical formula of two surfaces
after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other
Riemann–Hurwitz_formula
k_{2},\ldots ,k_{m}}={\frac {n!}{k_{1}!k_{2}!\dots k_{m}!}}} and the summation is taken over all sequences of nonnegative integer indices k1, k2, ..
Proofs of Fermat's little theorem
Proofs_of_Fermat's_little_theorem
Three results related to the density of prime numbers
{\displaystyle \sum _{p\leq n}{\frac {\log p}{p}}=\log n+O(1)} . A partial summation yields ∑ p ≤ n 1 p = log log n + M + O ( 1 / log n ) {\displaystyle
Mertens'_theorems
Conditions for switching order of integration in calculus
through results such as Cavalieri's principle, which was used by Leonhard Euler. More formally, the theorem states that if a function is Lebesgue integrable
Fubini's_theorem
Formula for area of a grid polygon
different way) as the basis for a proof of Euler's formula. Alternative proofs of Pick's theorem that do not use Euler's formula include the following. One can
Pick's_theorem
Numbers with a certain property involving recursive summation
Other polynomial numbers Hilbert Idoneal Leyland Loeschian Lucky numbers of Euler Williams Recursively defined numbers Fibonacci Jacobsthal Leonardo Lucas
Happy_number
Type of complex function with growth bounded by an exponential function
convergent summations over a series of other complex functions, as well as understanding when it is possible to apply techniques such as Borel summation, or
Exponential_type
Physics theorem for symmetries of action
{\textstyle \left|I\right|=k} . The summation convention does not directly apply to multiindices since the summation over lengths needs to be displayed
Noether's_second_theorem
On vector derivatives for rotating frames
trajectories in Lie group theory. Applying the product rule with implict summation convention, f ′ = ( f i b i ) ′ = ( f i T ) ′ e i = ( f i ′ T + f i G
Transport_theorem
Mathematical technique for improving convergence
example, to obtain a variety of identities on special functions. Thus, the Euler transform applied to the hypergeometric series gives some of the classic
Series_acceleration
Number of ways to pair up n objects
of the recurrence. The telephone numbers may be expressed exactly as a summation T ( n ) = ∑ k = 0 ⌊ n / 2 ⌋ ( n 2 k ) ( 2 k − 1 ) ! ! = ∑ k = 0 ⌊ n /
Telephone number (mathematics)
Telephone_number_(mathematics)
Meromorphic function on the complex plane
Fundamental subclasses of L-functions were built on the work of Leonhard Euler (which is now known as the Riemann zeta function). Most notably, the mathematicians
L-function
Geometric model of the physical space
In 1760, Euler proved a theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures, known as Euler's theorem.
Three-dimensional_space
Branch of mechanics concerned with balance of forces in nonmoving systems
velocity. The application of the assumption of zero acceleration to the summation of moments acting on the system leads to M = I α = 0 {\displaystyle {\textbf
Statics
equation Hamiltonian mechanics Poisson bracket Electrostatics Poisson equation Euler–Poisson–Darboux equation Poisson–Boltzmann equation Screened Poisson equation
List of things named after Siméon Denis Poisson
List_of_things_named_after_Siméon_Denis_Poisson
f(z)} . Ordinary summation succeeds only for common ratios | r | < 1. {\displaystyle |r|<1.} Cesàro summation Abel summation Euler summation The series is
Divergent_geometric_series
whether a given system of ordinary differential equations can arise as the Euler–Lagrange equations for some Lagrangian function. There has been a great
Inverse problem for Lagrangian mechanics
Inverse_problem_for_Lagrangian_mechanics
Israeli mathematician
which has revolutionized the field of hypergeometric summation. In 2004, Zeilberger was awarded the Euler Medal; the citation refers to him as "a champion
Doron_Zeilberger
Sequence in computer science
a prefix sum is known as a partial sum of a series. Prefix summation or partial summation form linear operators on the vector spaces of finite or infinite
Prefix_sum
Equations of motion for viscous fluids
term—hence describing viscous flow. The Navier–Stokes equations generalize the Euler equations which only consider inviscid flow. The Navier–Stokes equations
Navier–Stokes_equations
Tensor index notation for tensor-based calculations
convenience, the Ricci calculus incorporates Einstein notation, which implies summation over indices repeated within a term and universal quantification over
Ricci_calculus
American mathematician (1895–1969)
on Euler polynomials as a solution to a particular difference equation.[2] Cox used generalized Euler polynomials and the generalized Boole summation formula
Elbert_Frank_Cox
non-linear differential equations, and are commonly presented in the form of Euler–Lagrange equations of motion. However, they can also be presented as a set
Geodesics as Hamiltonian flows
Geodesics_as_Hamiltonian_flows
Special mathematical function defined as sin(x)/x
x}}={\frac {1}{\Gamma (1+x)\Gamma (1-x)}}={\frac {1}{\Pi (x)\Pi (-x)}}.} Euler discovered that sin ( x ) x = ∏ n = 1 ∞ cos ( x 2 n ) , {\displaystyle
Sinc_function
Origin and evolution of the symbols used to write equations and formulas
∞ 1 n 2 {\textstyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}} . For summation, Euler used an enlarged form of the upright capital Greek letter sigma (Σ)
History of mathematical notation
History_of_mathematical_notation
Class of numerical techniques
similar maximum principle also holds for the continuous case. The SBP-SAT (summation by parts - simultaneous approximation term) method is a stable and accurate
Finite_difference_method
Approximation technique in integral calculus
and gives a lower Riemann sum or lower Darboux sum. All these Riemann summation methods are among the most basic ways to accomplish numerical integration
Riemann_sum
Linear operator acting on modular forms
as moderate growth at infinity; these conditions are preserved by the summation, and so Hecke operators preserve the space of modular forms of a given
Hecke_operator
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EULER SUMMATION
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