Sequences, series, summability Definition
Convergent sequence
Builds on Limit of a sequence
Definition
A real sequence is convergent if it converges to some real number . Here convergence is in the sense of the limit of a sequence.
Remarks
The sequences and are convergent, with limits and . The constant sequence converges to . A sequence that is not convergent is a divergent sequence.
Reading path
To understand this page from the ground up, read these 4 pages in order. The foundations come first.
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- Sequence definition
- Real sequence definition
- Limit of a sequence definition
- Convergent sequence definition
Where this leads
279 pages build on this one.
- 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
- Conditional convergence
- Leibniz formula for pi
- Integral of exp(-x) over the half line
- Pythagorean trigonometric identity
- Limit of a quotient of sequences
- log x is at most x - 1
- Inverse hyperbolic functions
- Integral of cosine
- Comparison test
- Root test
- Derivative of arctangent
- Harmonic numbers grow like log n
- First fundamental theorem of calculus
- 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
- Cauchy–Hadamard theorem
- A function with positive derivative is strictly increasing
- Derivatives of the hyperbolic functions
- Derivative of the logarithm
- Divergence part of the ratio test
- Power series of cosine
- Integration by parts
- Cauchy product of absolutely convergent series
- Boundedness theorem
- Arcsine
- Cauchy condensation test
- Mean value theorem
- Mertens' theorem on Cauchy products
- 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
- The gamma function extends the factorial
- e as the limit of (1 + 1/n)^n
- 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
- 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
- The exponential function is strictly increasing
- Radius of convergence of the binomial series
- Derivative of x^a
- e
- Hyperbolic tangent
- A series of nonnegative terms converges iff its partial sums are bounded
- 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
- The exponential series has infinite radius of convergence
- Integral of a power
- Darboux's theorem
- 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
- Arccosine
- 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
- Term-by-term differentiation
- Real exponentiation
- Derivative of cosine
- Logarithm of a power
- Dirichlet's test
- Improper integral of x^p on an unbounded interval
- Product of continuous functions is continuous
- 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
- Wallis product
- Dini's theorem
- Second fundamental theorem of calculus
- Absolutely convergent series converge
- Mercator series
- A function with nonnegative second derivative is convex
- Cauchy criterion for sequences
- p-series test
- The logarithm is strictly concave
- Weierstrass factorization theorem
- 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
- Convergence of the Neumann series for Volterra kernels
- Series in Banach spaces
- Mehta integral
- Weierstrass product for the gamma function
- Hardy inequality in Sobolev spaces
- Brent's method
- High indices theorem
- Runge's theorem
- Lévy–Steinitz theorem
- Stein's lemma
- Bertrand's test
- 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
- Reduction of an initial value problem to a Volterra equation
- Schlömilch's generalization of the condensation test
- Ramanujan's series for 1/pi
- Neumann series
- Tauber's theorem
- 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
- Log-convexity of the gamma function
- Abelian and Tauberian theorems
- Lüroth's theorem
- Carleman's inequality
- Mean value theorem for divided differences
- Blancmange curve
- Euler product for multiplicative functions
- Hardy's Tauberian theorem for Cesàro means
- Regular summability method
- Tauberian condition
- Raabe's test
- Uniform convergence of absolutely summable Fourier series
- Differentiation operator as inverse of integration
- Completeness of the real numbers
- Logarithmically convex function
- Cauchy formula for repeated integration
- Lidskii's trace formula for integral operators
- 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
- Fourier basis of L2 of the circle
- Jentzsch's theorem on positive kernels
- Mercer's theorem
- Error bound for the trapezoidal rule
- Kempner series
- Bohr–Mollerup theorem
- Volterra's function
- Iterated exponentiation
- Noncommutative geometry
- An infinite product converges iff its series of logarithms converges
- Quintic equation
- Silverman–Toeplitz theorem
- Hardy's inequality
- Maximum and minimum
- Descartes' rule of signs
- Trigonometric integrals
- Basel problem via Fourier series
- Abel's summation formula
- Unique solvability of Volterra equations of the second kind
- Fubini's theorem for series
- Laplace transform of the exponential function
- Abel–Dini theorem
- Decay of Fourier coefficients and smoothness
- Tannery's theorem
- Function of exponential order
- Barnes G-function
- Littlewood's Tauberian theorem
- Chudnovsky algorithm
- Runge's phenomenon
Further reading
- Web https://en.wikipedia.org/wiki/Limit_of_a_sequence, Real numbers
About this page
| Tag | 0AZ3R52593 |
|---|---|
| URI | m2://0AZ3R52593 |
| Kind | definition |
| Writing | W1 stated |
| MSC | 40A05 |
| Defines | Convergent sequence |
| Uses | Limit of a sequence |
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What this page needs for W2
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
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What links here
Used by
- Divergent sequence
- Convergent series
- A sequence converges iff its liminf and limsup agree
- Bolzano–Weierstrass theorem
- Convergent sequences are Cauchy
- Convergent sequences are bounded
- Cauchy criterion for sequences
- Regular summability method
- Convergence of an infinite product
- Kronecker's lemma
- Kojima–Schur theorem
- Tannery's theorem