Sequences, series, summability Definition
Limit of a sequence
Builds on Real sequence
Definition
Let be a real sequence and let be a real number. The sequence converges to if the following holds for each open interval that contains . There is a natural number such that for all . Then is a limit of , with the notation or as .
Remarks
The epsilon-N definition of the limit of a sequence is an equivalent form, with intervals .
A sequence has at most one limit, by the uniqueness of limits. So the notation is well defined.
The same definition works in each topological space, with open neighbourhoods of in place of open intervals.
In Mathlib, the limit is the statement Filter.Tendsto a Filter.atTop (nhds L).
Reading path
To understand this page from the ground up, read these 3 pages in order. The foundations come first.
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- Sequence definition
- Real sequence definition
- Limit of a sequence definition
Where this leads
374 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
- 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
- A function with nonnegative derivative is increasing
- 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
- log x is at most x - 1
- Inverse hyperbolic functions
- Integral of cosine
- Comparison test
- Pointwise convergence
- Root test
- Derivative of arctangent
- Harmonic numbers grow like log n
- First fundamental theorem of calculus
- Arctangent
- 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
- 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
- Epsilon-N definition of the limit of a sequence
- 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
- 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
- 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
- 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
- Dirichlet integral
- Taylor's theorem with the Lagrange remainder
- Thomae's function
- A sequence converges iff its liminf and limsup agree
- 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
- 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
- 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
- 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
- Divergent series
- The integral of a continuous function is an antiderivative
- Fundamental theorem of calculus
- Triangle inequality for integrals
- Weierstrass M-test
- Sine is 2π-periodic
- A function with bounded derivative is Lipschitz
- 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
- Convergent sequences are Cauchy
- Continuity of the sum of a uniformly convergent series of functions
- pi is greater than 3
- Limit of a sum of sequences
- 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
- 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
- Term test
- Continuous functions on a closed bounded interval are uniformly continuous
- Absolute convergence
- Irrationality of e
- Tangent half-angle substitution
- 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
- 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
- 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
- Arnold tongue
- 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
- Uniform convergence of series
- Convergence of the Neumann series for Volterra kernels
- Daniell integral
- Series in Banach spaces
- Mehta integral
- Limit set
- 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
- Stein's lemma
- Bertrand's test
- Gamma function as Mellin transform of the exponential
- Padé approximant
- Rational approximation
- Purely transcendental extension
- 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
- Reduction of an initial value problem to a Volterra equation
- Schlömilch's generalization of the condensation test
- Beta function
- Uniform convergence of Bernstein approximations
- Ramanujan's series for 1/pi
- Neumann series
- Tauber's theorem
- Series acceleration
- Difference quotient
- Weierstrass product
- Continuous functional calculus
- Kummer's test
- Quadratic convergence of Newton's method
- Failure of Aizerman's conjecture
- Frullani integral
- Darboux function
- Gauss's test
- Mittag-Leffler summation
- Log-convexity of the gamma function
- Baire function
- Abelian and Tauberian theorems
- Lüroth's theorem
- Carleman's inequality
- Mean value theorem for divided differences
- Blancmange curve
- Euler product for multiplicative functions
- Watson's lemma
- Hardy's Tauberian theorem for Cesàro means
- Regular summability method
- Tauberian condition
- Raabe's test
- Legendre duplication formula
- Uniform convergence of absolutely summable Fourier series
- Mahler's theorem
- Karamata's Tauberian theorem
- Differentiation operator as inverse of integration
- Euler's sine product
- Laplace transform of the error function
- Completeness of the real numbers
- Logarithmically convex function
- Cauchy formula for repeated integration
- Lidskii's trace formula for integral operators
- Nested radical
- Approximation of a kernel by degenerate kernels
- Unconditional convergence
- Reduction of a first-kind Volterra equation to the second kind
- Liouville–Neumann series
- Convergence of an infinite product
- Dvoretzky–Rogers theorem
- Interpolation error formula
- Inversion of rational transforms by partial fractions
- 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
- q-Pochhammer symbol
- Density of trigonometric polynomials in continuous functions
- Padé table
- 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
- Steinhaus theorem on regular methods
- Cantor function
- 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
- Mercer's theorem
- Error bound for the trapezoidal rule
- Korovkin's theorem
- Kempner series
- Bohr–Mollerup theorem
- Poles of the gamma function
- Moore–Osgood theorem
- Normal family
- Wynn's epsilon algorithm
- Aitken's delta-squared process
- Volterra's function
- Iterated exponentiation
- Laplace transform of a power function
- Noncommutative geometry
- An infinite product converges iff its series of logarithms converges
- Quintic equation
- Silverman–Toeplitz theorem
- Shanks transformation
- Hardy's inequality
- Maximum and minimum
- Hölder's theorem
- Euler–Gauss limit formula
- Descartes' rule of signs
- Trigonometric integrals
- Uniform approximation
- Fejér's theorem
- Basel problem via Fourier series
- Abel's summation formula
- Gamma function has no zeros
- Wielandt's theorem
- Reflection formula for the gamma function
- Unique solvability of Volterra equations of the second kind
- Gamma of one half
- Fubini's theorem for series
- Laplace transform of the exponential function
- Abel–Dini theorem
- Decay of Fourier coefficients and smoothness
- Tannery's theorem
- Gauss multiplication formula
- Function of exponential order
- Infinite compositions of functions
- Barnes G-function
- Reciprocal gamma function
- Automorphism group of a bounded domain is a Lie group
- Multiplication theorem
- Oscillation of a function
- Littlewood's Tauberian theorem
- Chudnovsky algorithm
- Euler integral for the gamma function
- Functional equation of the gamma function
- Holomorphy of the gamma function
- Runge's phenomenon
- Montel's theorem
Further reading
- Web https://en.wikipedia.org/wiki/Limit_of_a_sequence, Real numbers
- Web https://en.wikipedia.org/wiki/Limit_of_a_sequence, Topological spaces
About this page
| Tag | 0XKWSSY8P2 |
|---|---|
| URI | m2://0XKWSSY8P2 |
| Kind | definition |
| Writing | W1 stated |
| MSC | 40A05 |
| Defines | Limit of a sequence |
| Uses | Real sequence |
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What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Used by
- Sequential characterization of limits
- Limit of a quotient of sequences
- Pointwise convergence
- Limit of a product of sequences
- Algebra of limits
- The sequence 1/n tends to zero
- Epsilon-N definition of the limit of a sequence
- The nth root of n tends to 1
- Convergent sequence
- Monotone convergence theorem for sequences
- Limit comparison test
- Sequential continuity
- Exponentials dominate powers in sequences
- Limits preserve non-strict inequalities
- Stolz–Cesàro theorem
- The sequence r^n tends to zero for |r| < 1
- Limit of a sum of sequences
- Cesàro means of a convergent sequence converge
- Subsequences of a convergent sequence converge to the same limit
- Squeeze theorem
- Uniqueness of limits
- Limit set
- Series acceleration
- Infinite compositions of functions