Sequences, series, summability Definition
Absolute convergence
Builds on SeriesConvergent series
Definition
Let be a real sequence. The series converges absolutely if the series of absolute values converges.
Remarks
The series has nonnegative terms. So it converges exactly when its partial sums are bounded. Absolutely convergent series converge, but the converse fails. The alternating harmonic series converges, but not absolutely. A convergent series that does not converge absolutely is conditionally convergent. The same definition works for complex terms, with the modulus in place of the absolute value.
Reading path
To understand this page from the ground up, read these 8 pages in order. The foundations come first.
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- Sequence definition
- Real sequence definition
- Limit of a sequence definition
- Series definition
- Convergent sequence definition
- Partial sum definition
- Convergent series definition
- Absolute convergence definition
Where this leads
117 pages build on this one.
- Power series are differentiable inside the radius of convergence
- Natural logarithm
- Euler–Mascheroni constant
- The logarithm grows slower than any power
- Derivative of sine
- 1 + x is at most exp x
- cos pi = -1
- sin x / x tends to 1
- Angle addition formulas
- Inverse trigonometric functions
- Power series of sine
- Conditional convergence
- Leibniz formula for pi
- Integral of exp(-x) over the half line
- Pythagorean trigonometric identity
- log x is at most x - 1
- Inverse hyperbolic functions
- Integral of cosine
- Root test
- Derivative of arctangent
- Harmonic numbers grow like log n
- Integral of the exponential function
- Power series of the exponential function
- cosh^2 - sinh^2 = 1
- Cauchy–Hadamard theorem
- Derivatives of the hyperbolic functions
- Derivative of the logarithm
- Divergence part of the ratio test
- Power series of cosine
- Cauchy product of absolutely convergent series
- Arcsine
- Mertens' theorem on Cauchy products
- pi
- Exponential growth beats polynomial growth
- The exponential function is convex
- Gaussian integral
- Basel problem
- Riemann rearrangement theorem
- 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
- Sum of the reciprocals of the factorials
- Derivative of arcsine
- Addition formulas for hyperbolic functions
- Hyperbolic sine
- Integral of sine
- Hyperbolic functions
- Rearrangements of an absolutely convergent series have the same sum
- Logarithm of a quotient
- Ratio formula for the radius of convergence
- Integral of 1/x
- Logarithm of a product
- Stirling's formula
- The exponential function is strictly increasing
- Radius of convergence of the binomial series
- Derivative of x^a
- e
- Hyperbolic tangent
- Triangle inequality for integrals
- Sine is 2π-periodic
- The exponential function is positive
- The exponential series has infinite radius of convergence
- Irrationality of pi
- Ratio test
- pi is greater than 3
- Arccosine
- Logarithm is the inverse of the exponential
- Exponential function
- Power-reduction formula for cosine
- Derivative of tangent
- Power series of arctangent
- Real exponentiation
- Derivative of cosine
- Logarithm of a power
- Improper integral of x^p on an unbounded interval
- Irrationality of e
- Integration by partial fractions
- Absolutely convergent series converge
- Mercator series
- The logarithm is strictly concave
- Weierstrass factorization theorem
- Riemann zeta function as a series
- Mehta integral
- Weierstrass product for the gamma function
- Lévy–Steinitz theorem
- Bertrand's test
- The exponential map from the additive reals to the positive reals
- Mertens' theorems
- Eisenstein series as a double series
- Ramanujan's series for 1/pi
- Weierstrass product
- Kummer's test
- Frullani integral
- Gauss's test
- Log-convexity of the gamma function
- Euler product for multiplicative functions
- Raabe's test
- Logarithmically convex function
- Dvoretzky–Rogers theorem
- Laplace transforms of sine and cosine
- Hadamard factorization theorem
- Viète's formula
- Interchange of summation order
- Bohr–Mollerup theorem
- Iterated exponentiation
- An infinite product converges iff its series of logarithms converges
- Trigonometric integrals
- Basel problem via Fourier series
- Fubini's theorem for series
- Laplace transform of the exponential function
- Tannery's theorem
- Function of exponential order
- Barnes G-function
- Chudnovsky algorithm
Further reading
- Web https://en.wikipedia.org/wiki/Absolute_convergence, Definition for real and complex numbers
About this page
| Tag | 0W3K3QPEWT |
|---|---|
| URI | m2://0W3K3QPEWT |
| Aliases | absolutely-convergent-series |
| Kind | definition |
| Writing | W1 stated |
| MSC | 40A05 |
| Defines | Absolute convergence |
| Uses | Series, Convergent series |
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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
- Conditional convergence
- Root test
- Cauchy product of absolutely convergent series
- Mertens' theorem on Cauchy products
- Rearrangements of an absolutely convergent series have the same sum
- Triangle inequality for integrals
- Ratio test
- Absolutely convergent series converge
- Riemann zeta function as a series
- Lévy–Steinitz theorem
- Eisenstein series as a double series
- Euler product for multiplicative functions
- Dvoretzky–Rogers theorem
- Interchange of summation order
- Fubini's theorem for series
- Tannery's theorem