Sequences, series, summability Definition
Bounded sequence
Builds on Real sequence
Definition
Let be a real sequence. The sequence is bounded above if there is a real number with for all . The sequence is bounded below if there is a real number with for all . The sequence is bounded if it is bounded above and bounded below. Equivalently, is bounded if there is a real number with for all .
Remarks
The sequence is bounded, with . The sequence is bounded below by , but it is not bounded above. A sequence is a function on , so a bounded sequence is a bounded function on . Each convergent sequence is bounded, as the node convergent sequences are bounded states.
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
- Bounded sequence definition
Where this leads
314 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
- Riemann's criterion for integrability
- Rolle's theorem
- Integral test
- Weierstrass function
- First derivative test
- cos pi = -1
- sin x / x tends to 1
- Uniformly Cauchy sequence of functions
- Darboux and Riemann integrals agree
- A function with nonnegative derivative is increasing
- Inverse trigonometric functions
- Power series of sine
- Cauchy mean value theorem
- Decreasing sequence bounded below converges
- 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
- Monotone functions are Riemann integrable
- Comparison test
- 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
- 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
- Divergence part of the ratio test
- Power series of cosine
- Integration by parts
- Upper and lower Darboux integrals
- Boundedness theorem
- Arcsine
- Cauchy condensation test
- Mean value theorem
- 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
- Integration by substitution
- The exponential function is convex
- Gaussian integral
- Basel problem
- Exponential of a sum
- 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
- 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
- Dirichlet integral
- Limit inferior
- 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
- Logarithm of a quotient
- Ratio formula for the radius of convergence
- Alternating harmonic series
- Weierstrass approximation theorem
- L'Hôpital's rule
- 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
- e
- Hyperbolic tangent
- A series of nonnegative terms converges iff its partial sums are bounded
- Mean value theorem for integrals
- The integral of a continuous function is an antiderivative
- Fundamental theorem of calculus
- Weierstrass M-test
- Sine is 2π-periodic
- A function with bounded derivative is Lipschitz
- Limit superior
- The exponential function is positive
- Alternating series test
- The exponential series has infinite radius of convergence
- Integral of a power
- Darboux's theorem
- Irrationality of pi
- Ratio test
- Upper and lower Darboux sums
- 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
- Convergent sequences are bounded
- Derivative of tangent
- Normal convergence
- Power series of arctangent
- 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
- Continuous functions on a closed bounded interval are uniformly continuous
- 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
- Wallis product
- Dini's theorem
- Second fundamental theorem of calculus
- Absolutely convergent series converge
- Term-by-term integration
- Mercator series
- A function with nonnegative second derivative is convex
- Interval of convergence
- Cauchy criterion for sequences
- p-series test
- The logarithm is strictly concave
- 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
- Riemann zeta function as a series
- Stein's unbiased risk estimate
- L2 convergence of Fourier series
- Convergence of the Neumann series for Volterra kernels
- Hardy's uncertainty principle
- Scheffé's lemma
- Mehta integral
- Smoothing by convolution
- Darboux integral
- Stirling's formula for the gamma function
- Weierstrass product for the gamma function
- Brent's method
- High indices theorem
- Log-gamma function
- Runge's theorem
- Stein's lemma
- Bertrand's test
- Gamma function as Mellin transform of the exponential
- Padé approximant
- Rational approximation
- Purely transcendental extension
- Mittag-Leffler star
- The exponential map from the additive reals to the positive reals
- Mertens' theorems
- Peano kernel theorem
- Stein's example
- Bisection method
- Reduction of an initial value problem to a Volterra equation
- Schlömilch's generalization of the condensation test
- Beta function
- Ramanujan's series for 1/pi
- Cantor–Lebesgue theorem
- Difference quotient
- Weierstrass product
- Continuous functional calculus
- Derivative of a convolution
- 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
- Lüroth's theorem
- Mean value theorem for divided differences
- Blancmange curve
- Watson's lemma
- Dominated convergence theorem
- Vitali convergence theorem
- Covering density
- Entropic uncertainty principle
- Raabe's test
- Legendre duplication formula
- Uniform convergence of absolutely summable Fourier series
- Hadamard gap theorem
- Karamata's Tauberian theorem
- Differentiation operator as inverse of integration
- Laplace transform of the error function
- Completeness of the real numbers
- Logarithmically convex function
- 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
- Fabry gap theorem
- Reduction of a first-kind Volterra equation to the second kind
- Liouville–Neumann series
- Interpolation error formula
- Differentiation and multiplication under the Fourier transform
- Inversion of rational transforms by partial fractions
- Nevanlinna theory
- Hilbert's seventeenth problem
- Laplace transforms of sine and cosine
- Newman's theorem on rational approximation of the absolute value
- Density of trigonometric polynomials in continuous functions
- Padé table
- Hadamard factorization theorem
- Borel's lemma
- Viète's formula
- Heaviside cover-up method
- Nested intervals theorem
- Compactness of integral operators
- Laplace transform tends to zero at infinity
- Steinhaus theorem on regular methods
- 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
- Particular values of the gamma function
- Mercer's theorem
- Error bound for the trapezoidal rule
- Bohr–Mollerup theorem
- Poles of the gamma function
- Volterra's function
- Iterated exponentiation
- Laplace transform of a power function
- Borel summation
- Noncommutative geometry
- An infinite product converges iff its series of logarithms converges
- Quintic equation
- Fatou's lemma
- Maximum and minimum
- Hölder's theorem
- Euler–Gauss limit formula
- Descartes' rule of signs
- Trigonometric integrals
- Basel problem via Fourier series
- 1 − 1 + 2 − 6 + 24 − 120 + ⋯
- Abel's summation formula
- Exponential type
- 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
- Laplace transform of the exponential function
- Abel–Dini theorem
- Decay of Fourier coefficients and smoothness
- Gauss multiplication formula
- Function of exponential order
- Semi-continuity
- Dini derivative
- Barnes G-function
- Mazur–Orlicz theorem
- Reciprocal gamma function
- Multiplication theorem
- Order of an entire function
- 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
Further reading
- Web https://en.wikipedia.org/wiki/Sequence, Bounded
- Web https://en.wikipedia.org/wiki/Bounded_function, lead section
About this page
| Tag | 05FQ3VGAR9 |
|---|---|
| URI | m2://05FQ3VGAR9 |
| Kind | definition |
| Writing | W0 stub |
| MSC | 40A05 |
| Defines | Bounded sequence |
| Uses | Real sequence |
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