# absolute convergence

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Related to absolutely convergent: Conditionally convergent

## absolute convergence

[′ab·sə‚lüt kən′vərj·əns]
(mathematics)
That property of an infinite series (or infinite product) of real or complex numbers if the series (product) of absolute values converges; absolute convergence implies convergence.
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References in periodicals archive ?
In this region we have to take care of the order of the Euler factors participating in the Euler product, since it is not absolutely convergent.
It is easily seen the Dirichlet series is absolutely convergent for Rs = [sigma] > 1/6.
rho]([absolute value of y'])+[alpha]]] (x) is absolutely convergent.
k,l] is absolutely convergent if and only if the following conditions hold: (i) There are ([k.
is absolutely convergent for [sigma] < O, since the term in square brackets vanishes for 0 [less than or equal to] x < 1 by (11).
0] of D is connected semi-simple, then I'([phi]) is absolutely convergent for every [phi][Epsilon] S(XA), and this convergence is invariant under castling transforms, which means that if two irreducible regular prehomogeneous vector spaces (G,X) and [Mathematical Expressions Omitted] are castling transforms of each other and [Mathematical Expressions Omitted] for (G,X) is absolutely convergent for every [phi][Epsilon] S([X.
It is easily seen that H(s) can be written as Dirichlet series which is absolutely convergent for [Real part]s > 1/6.
where the last infinite sum is absolutely convergent if [absolute value of u] is much smaller than [[absolute value of t].
K] is defined by the following absolutely convergent Euler product:
where G(s) can be written as a Dirichlet series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is absolutely convergent for Rs [greater than or equal to] 1/5.
n=1] is absolutely convergent for Res [greater than or equal to] - 2/5, so we have
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] > 0 and the Dirichlet series M(s) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is absolutely convergent for Rs > 1/5.

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