Absolute convergence

The absolute convergence is a term from the analysis and is used in connection with rows. For the absolutely convergent series, some properties of finite sums remain valid that are wrong for the greater amount of convergent series in general.

Definition

• A real-valued or complex- number is called absolutely convergent if the series of absolute values ​​converges.

• This definition is generalized to normed spaces. A series in a normed space is called absolutely convergent if the series of norms converges.

Examples

• Convergent series whose summands are almost all non-negative, are absolutely convergent.
• The series is absolutely convergent for.

• The power series of the exponential function is absolutely convergent for every complex.
• The general rule is that a real or complex power series inside its circle of convergence is absolutely convergent.
• The alternating harmonic series is convergent against. But it is not absolutely convergent, because when verifying the defining characteristic is obtained, so the ordinary harmonic series. This is definitely divergent against.

Properties

Every absolutely convergent series is convergent. This applies to both real-valued and complex-valued for rows. Since it vice versa are rows that are convergent but not absolutely convergent ( such series are conditionally convergent called ), is the set of absolutely convergent series is a proper subset of the set of convergent series.

Some convergence criteria for rows to prove the absolute convergence. These include the root test and the ratio test.

Rearrangements

An essential property of absolutely convergent series is that you can swap any summands as finite sums: Any rearrangement of an absolutely convergent series, ie each row is formed by rearrangement of the number of members is convergent and converges to the same threshold value as. This is exactly the reverse of convergent but not absolutely convergent series: There always exists a rearrangement of which diverges.

If the number is real-valued, the following sharper statement ( Riemann rearrangement theorem ) applies: At any given number, there is a rearrangement of the series that converges to (improperly ). The reason is easy to specify, we restrict ourselves to the case. It maps the summand in two episodes

To ( addends are equal to zero, can be omitted ). Now you add as long followers of until is exceeded, then the (negative) terms of the sequence, until has fallen again, then again, etc. The method is practical because and diverge (otherwise would be the original series absolutely convergent ), and the rearranged series converges to.

Generalizations

The concept of absolute convergence can be generalized to normed spaces. Given is a sequence of elements of a normed space. The corresponding series is

Defined. The series is called absolutely convergent if converges.

If a Banach space, so complete, so every absolutely convergent series is convergent. Indeed thereof converse is also true: If a normed vector space and each absolutely convergent series is convergent, so is complete, ie it is a Banach space.

In any complete metric spaces, a related result holds. A sequence at least convergent if the sum

Converges. As in the example above, yes, the absolute convergence follows from it as a special case.