Sequences, series, summability Definition
Real sequence
Builds on Sequence
Definition
A real sequence is a sequence in the set of real numbers. Thus it is a function , and each term is a real number.
Remarks
Examples are , and . The real sequences form a real vector space under the termwise operations and . The limit of a sequence is defined first for real sequences in most courses.
Reading path
To understand this page from the ground up, read these 2 pages in order. The foundations come first.
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- Sequence definition
- Real sequence definition
Where this leads
492 pages build on this one.
- Sequential characterization of limits
- Power series are differentiable inside the radius of convergence
- A function with zero derivative is constant
- Natural logarithm
- A differentiable function is convex iff its derivative is increasing
- Taylor's theorem with the Cauchy remainder
- Euler–Mascheroni constant
- The logarithm grows slower than any power
- Derivative of sine
- 1 + x is at most exp x
- Reduction formula
- Alternating series
- Riemann's criterion for integrability
- Rolle's theorem
- Integral test
- Weierstrass function
- First derivative test
- cos pi = -1
- Divergent sequence
- sin x / x tends to 1
- Geometric series
- Uniformly Cauchy sequence of functions
- Darboux and Riemann integrals agree
- Taylor series
- A function with nonnegative derivative is increasing
- Uniformly continuous functions map Cauchy sequences to Cauchy sequences
- Angle addition formulas
- Inverse trigonometric functions
- Power series of sine
- Cauchy mean value theorem
- Decreasing sequence bounded below converges
- Conditional convergence
- Leibniz formula for pi
- Integral of exp(-x) over the half line
- Pythagorean trigonometric identity
- Limit of a quotient of sequences
- The Dirichlet function is not Riemann integrable
- Maclaurin series
- log x is at most x - 1
- Inverse hyperbolic functions
- Integral of cosine
- Monotone functions are Riemann integrable
- Comparison test
- Pointwise convergence
- Sum of a finite geometric series
- Bounded sequence
- Root test
- Derivative of arctangent
- Harmonic numbers grow like log n
- First fundamental theorem of calculus
- Arctangent
- Radius of convergence
- Integral of the exponential function
- Power series of the exponential function
- Cauchy criterion for series
- cosh^2 - sinh^2 = 1
- Continuous functions are Riemann integrable
- Tangent function
- Continuous image of a closed interval is a closed interval
- Limit of a product of sequences
- Continuity of the inverse of a strictly monotone continuous function
- Cauchy–Hadamard theorem
- A function with positive derivative is strictly increasing
- Derivatives of the hyperbolic functions
- Derivative of the logarithm
- Algebra of limits
- Divergence part of the ratio test
- Uniform convergence
- Uniform convergence on a set
- The sequence 1/n tends to zero
- Power series of cosine
- Integration by parts
- Sequence tending to infinity
- Upper and lower Darboux integrals
- Epsilon-N definition of the limit of a sequence
- Smooth non-analytic function
- sin(1/x) has no limit at 0
- Cauchy product of absolutely convergent series
- Boundedness theorem
- The nth root of n tends to 1
- Convergent sequence
- Cauchy product
- Arcsine
- Cauchy condensation test
- Mean value theorem
- Mertens' theorem on Cauchy products
- Uniform limits commute with integration
- Power series converge uniformly on smaller discs inside the radius of convergence
- pi
- Polynomial functions are continuous
- Abel's test
- Uniform convergence of derivatives allows term-by-term differentiation
- Exponential growth beats polynomial growth
- Convergent series
- Harmonic series
- Contractive sequence
- Integration by substitution
- The exponential function is convex
- Gaussian integral
- Basel problem
- Riemann rearrangement theorem
- Exponential of a sum
- Harmonic series diverges
- Trigonometric substitution
- Hyperbolic cosine
- The exponential function is its own derivative
- x sin(1/x) is continuous at 0
- The gamma function extends the factorial
- e as the limit of (1 + 1/n)^n
- Monotone convergence theorem for sequences
- Comparison test for improper integrals
- Second derivative test
- Sum of the reciprocals of the factorials
- Cauchy sequences are bounded
- Derivative of arcsine
- Addition formulas for hyperbolic functions
- Quotient of continuous functions is continuous
- Decimal expansion as a series
- Hyperbolic sine
- Area under a curve
- Integral of sine
- Telescoping series
- Dirichlet integral
- Limit inferior
- Taylor's theorem with the Lagrange remainder
- Binomial series
- Thomae's function
- A sequence converges iff its liminf and limsup agree
- Rearrangement of a series
- Limit comparison test
- Existence of nth roots by the intermediate value theorem
- Hyperbolic functions
- Rearrangements of an absolutely convergent series have the same sum
- Logarithm of a quotient
- Ratio formula for the radius of convergence
- Analytic function
- Series
- Alternating harmonic series
- Weierstrass approximation theorem
- L'Hôpital's rule
- Sum of a series
- Integral of 1/x
- Logarithm of a product
- Stirling's formula
- Bolzano–Weierstrass theorem
- One-sided limits of a monotone function exist
- Monotone sequence
- The exponential function is strictly increasing
- Radius of convergence of the binomial series
- Derivative of x^a
- Derivative of an inverse function
- Linearity of differentiation
- Derivative of a polynomial
- Every sequence has a monotone subsequence
- Geometric sequence
- Series of functions
- e
- Sequential continuity
- Exponentials dominate powers in sequences
- Hyperbolic tangent
- A series of nonnegative terms converges iff its partial sums are bounded
- Limits preserve non-strict inequalities
- Mean value theorem for integrals
- Partial sum
- Divergent series
- The integral of a continuous function is an antiderivative
- Fundamental theorem of calculus
- Triangle inequality for integrals
- Weierstrass M-test
- Cesàro mean
- Sine is 2π-periodic
- A function with bounded derivative is Lipschitz
- Limit superior
- The exponential function is positive
- Continuity is equivalent to sequential continuity
- Alternating series test
- The exponential series has infinite radius of convergence
- Stolz–Cesàro theorem
- Integral of a power
- Darboux's theorem
- The sequence r^n tends to zero for |r| < 1
- Irrationality of pi
- Ratio test
- Upper and lower Darboux sums
- Convergent sequences are Cauchy
- Cauchy sequence
- Continuity of the sum of a uniformly convergent series of functions
- pi is greater than 3
- Limit of a sum of sequences
- Cluster point of a sequence
- Arccosine
- Cesàro means of a convergent sequence converge
- Logarithm is the inverse of the exponential
- Exponential function
- A continuous nowhere differentiable function
- Power-reduction formula for cosine
- Abel's theorem
- Convergent sequences are bounded
- Derivative of tangent
- Normal convergence
- Power series of arctangent
- Dirichlet function
- Term-by-term differentiation
- Real exponentiation
- The sequence r^n tends to infinity for r > 1
- Derivative of cosine
- Gamma function
- Logarithm of a power
- Dirichlet's test
- Improper integral of x^p on an unbounded interval
- Product of continuous functions is continuous
- Subsequences of a convergent sequence converge to the same limit
- Power series
- Term test
- Continuous functions on a closed bounded interval are uniformly continuous
- Absolute convergence
- Irrationality of e
- Tangent half-angle substitution
- Sine and cosine
- Integration by partial fractions
- Rational function
- x^2 sin(1/x) is differentiable with discontinuous derivative
- Taylor's theorem with the integral remainder
- Extreme value theorem
- A monotone function has countably many discontinuities
- Uniform limit theorem
- Wallis product
- Limit of a sequence
- Squeeze theorem
- Dini's theorem
- Second fundamental theorem of calculus
- Absolutely convergent series converge
- Sequence of functions
- Term-by-term integration
- x^n converges pointwise but not uniformly on [0, 1]
- Mercator series
- A function with nonnegative second derivative is convex
- Interval of convergence
- Constant multiple rule
- Cauchy criterion for sequences
- p-series test
- The logarithm is strictly concave
- Uniqueness of limits
- Weierstrass factorization theorem
- Beta function in terms of the gamma function
- Abelian theorems for integral transforms
- Arnold tongue
- Fourier transform on Schwartz space
- Stone–Weierstrass theorem
- Lebesgue's theorem on differentiability of monotone functions
- Minimal curvature property of natural cubic splines
- Köthe–Toeplitz dual
- Riemann zeta function as a series
- Stein's unbiased risk estimate
- L2 convergence of Fourier series
- Fejér kernel
- Truncation error
- Uniform convergence of series
- Convergence of the Neumann series for Volterra kernels
- Daniell integral
- Hardy's uncertainty principle
- Nörlund mean
- Scheffé's lemma
- Series in Banach spaces
- Mehta integral
- Smoothing by convolution
- Limit set
- Darboux integral
- Stirling's formula for the gamma function
- Weierstrass product for the gamma function
- Hardy inequality in Sobolev spaces
- Strongly measurable function
- Brent's method
- Mergelyan's theorem
- High indices theorem
- Log-gamma function
- Runge's theorem
- Lévy–Steinitz theorem
- Bump function
- Stein's lemma
- Bertrand's test
- Cesàro summation
- Gamma function as Mellin transform of the exponential
- Padé approximant
- Rational approximation
- Euler transform
- Purely transcendental extension
- Mittag-Leffler star
- The exponential map from the additive reals to the positive reals
- Mertens' theorems
- Peano kernel theorem
- Eisenstein series as a double series
- Stein's example
- Bisection method
- Levin-type transformations
- Kingman's subadditive ergodic theorem
- Reduction of an initial value problem to a Volterra equation
- Fekete's lemma
- Schlömilch's generalization of the condensation test
- Beta function
- Quasi-analytic function
- Uniform convergence of Bernstein approximations
- Ramanujan's series for 1/pi
- Neumann series
- Tauber's theorem
- Cantor–Lebesgue theorem
- Series acceleration
- Difference quotient
- Weierstrass product
- Continuous functional calculus
- Derivative of a convolution
- Kummer's test
- Euler–Maclaurin formula
- Quadratic convergence of Newton's method
- Constructive analysis
- Failure of Aizerman's conjecture
- Frullani integral
- Darboux function
- Gauss's test
- Mittag-Leffler summation
- Van Wijngaarden transformation
- Log-convexity of the gamma function
- Baire function
- Abelian and Tauberian theorems
- Lüroth's theorem
- Analytic function between Banach spaces
- Carleman's inequality
- Consistency of summability methods
- Mean value theorem for divided differences
- Blancmange curve
- Strong Cesàro summability
- Euler product for multiplicative functions
- Watson's lemma
- Hardy's Tauberian theorem for Cesàro means
- Regular summability method
- Dominated convergence theorem
- Tauberian condition
- Vitali convergence theorem
- Covering density
- Entropic uncertainty principle
- Raabe's test
- Trigonometric functions
- Legendre duplication formula
- Smoothness classes
- Uniform convergence of absolutely summable Fourier series
- Mahler's theorem
- Hadamard gap theorem
- Karamata's Tauberian theorem
- Differentiation operator as inverse of integration
- Euler's sine product
- Laplace transform of the error function
- Newton series
- Completeness of the real numbers
- Logarithmically convex function
- Hartogs's theorem on separate holomorphicity
- Heisenberg inequality for the Fourier transform
- Cauchy formula for repeated integration
- Lidskii's trace formula for integral operators
- Nested radical
- Babenko–Beckner inequality
- Approximation of a kernel by degenerate kernels
- Complex p-adic numbers
- Krasner's lemma
- Unconditional convergence
- Fabry gap theorem
- Reduction of a first-kind Volterra equation to the second kind
- Space of convergent sequences
- Liouville–Neumann series
- Convergence of an infinite product
- Dvoretzky–Rogers theorem
- Interpolation error formula
- Differentiation and multiplication under the Fourier transform
- Inversion of rational transforms by partial fractions
- Nevanlinna theory
- Kronecker's lemma
- Hilbert's seventeenth problem
- Kojima–Schur theorem
- Laplace transforms of sine and cosine
- Newman's theorem on rational approximation of the absolute value
- Berkovich space dynamics
- Wiman–Valiron theory
- q-Pochhammer symbol
- Density of trigonometric polynomials in continuous functions
- Padé table
- Completion of a valued field
- Denjoy–Carleman theorem
- Spectral radius formula
- Hadamard factorization theorem
- Borel's lemma
- Viète's formula
- Heaviside cover-up method
- Convergence tests
- Nested intervals theorem
- Compactness of integral operators
- Interchange of summation order
- Laplace transform tends to zero at infinity
- Hardy–Littlewood Tauberian theorem
- Osgood's lemma
- Lindelöf summation
- Steinhaus theorem on regular methods
- Oseledets theorem
- Cantor function
- Cantor's uniqueness theorem for trigonometric series
- Commutative Gelfand–Naimark theorem
- Tauberian theorems for integral transforms
- Fourier basis of L2 of the circle
- Jentzsch's theorem on positive kernels
- Laguerre–Pólya class
- Particular values of the gamma function
- Absolute summability
- Mercer's theorem
- Wilf–Zeilberger pair
- Error bound for the trapezoidal rule
- Korovkin's theorem
- Kempner series
- Riesz mean
- Bohr–Mollerup theorem
- Poles of the gamma function
- Moore–Osgood theorem
- Normal family
- Wynn's epsilon algorithm
- Aitken's delta-squared process
- Frobenius theorem on Cesàro and Abel summability
- Volterra's function
- Iterated exponentiation
- Laplace transform of a power function
- Matrix summability method
- Borel summation
- Zeilberger's algorithm
- Noncommutative geometry
- An infinite product converges iff its series of logarithms converges
- Quintic equation
- Silverman–Toeplitz theorem
- Shanks transformation
- Hardy's inequality
- Chebyshev's sum inequality
- Fatou's lemma
- Maximum and minimum
- Hölder's theorem
- Euler–Gauss limit formula
- Rate of convergence
- Descartes' rule of signs
- Trigonometric integrals
- Hölder summation
- Uniform approximation
- Fejér's theorem
- Basel problem via Fourier series
- 1 − 1 + 2 − 6 + 24 − 120 + ⋯
- Abel's summation formula
- Trigonometric definition of Chebyshev polynomials
- Pesin theory
- Exponential type
- Euler summation
- Gamma function has no zeros
- Wielandt's theorem
- Space of null sequences
- Reflection formula for the gamma function
- Unique solvability of Volterra equations of the second kind
- Gamma of one half
- Antithetic variates
- Linear and stable summation methods
- Fubini's theorem for series
- Laplace transform of the exponential function
- Abel–Dini theorem
- Summability method
- Decay of Fourier coefficients and smoothness
- Tannery's theorem
- Gauss multiplication formula
- Function of exponential order
- Semi-continuity
- Infinite compositions of functions
- Euler's continued fraction formula
- Dini derivative
- Barnes G-function
- Mazur–Orlicz theorem
- Chebyshev constant
- Reciprocal gamma function
- Automorphism group of a bounded domain is a Lie group
- Multiplication theorem
- Oscillation of a function
- Order of an entire function
- Lambert summation
- Abel summation
- Littlewood's Tauberian theorem
- Chudnovsky algorithm
- Euler integral for the gamma function
- Functional equation of the gamma function
- Hartogs lemma for subharmonic functions
- Holomorphy of the gamma function
- Runge's phenomenon
- Fourier transform of the Gaussian
- Montel's theorem
Further reading
- Web https://en.wikipedia.org/wiki/Sequence, Limits and convergence
About this page
| Tag | 0CTV783HYT |
|---|---|
| URI | m2://0CTV783HYT |
| Kind | definition |
| Writing | W0 stub |
| MSC | 40A05 |
| Defines | Real sequence |
| Uses | Sequence |
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