absolute convergence

The series converges, as well, by the Absolute convergence test.

The line and half-plane of absolute convergence can be defined similarly.

(this is the only place where the absolute convergence is used).

See more about absolute convergence of Fourier series.

The notion of absolute convergence plays a central role in the theory of infinite series.

When G is complete, absolute convergence implies unconditional convergence.

Sometimes the notion of absolute convergence does enter the discussion, however, especially in the study of the convergence problem.

In particular, for series with values in any Banach space, absolute convergence implies convergence.

(For issues with infinite summation, see absolute convergence.)

The Dirichlet series case is more complicated, though: absolute convergence and uniform convergence may occur in distinct half-planes.

Alternatively, the test may be stated in terms of absolute convergence, in which case it also applies to series with complex terms:

Thus, there might be a strip between the line of convergence and absolute convergence where a Dirichlet series is conditionally convergent.

For real series it follows from the Riemann rearrangement theorem that unconditional convergence implies absolute convergence.

In general, the abscissa of convergence does not coincide with abscissa of absolute convergence.

In a normed vector space, one can define absolute convergence as convergence of the series of norms ().

"Pointwise absolute convergence" is then simply pointwise convergence of .

Also Cauchy's well-known test for absolute convergence stems from this book: Cauchy condensation test.

Here μ is the Möbius mu function, and the rearrangement of terms is justified by absolute convergence.

Many standard tests for divergence and convergence, most notably including the ratio test and the root test, demonstrate absolute convergence.

The converse is also true: if absolute convergence implies convergence in a normed space, then the space is a Banach space.

For functions taking values in a normed linear space, absolute convergence refers to convergence of the series of positive, real-valued functions .

Intuitively, this is because the absolute-convergence gets slower and slower as x approaches 1, where convergence holds but absolute convergence fails.

In other words, for absolute convergence there is no issue of where the sum converges absolutely - if it converges absolutely at one point then it does so everywhere.

Since a Dirichlet character χ is completely multiplicative, its L-function can also be written as an Euler product in the half-plane of absolute convergence:

The notion of absolute convergence requires more structure, namely a norm, which is a real-valued function on abelian group G (written additively, with identity element 0) such that: