Riemann series theorem

The Riemann rearrangement theorem ( after Bernhard Riemann ) is a mathematical theorem about conditionally convergent series.

Formulation

Is a conditionally convergent series of real numbers, there exists at any arbitrary predetermined real number, a rearrangement of the series members so that the reordered series converges. For there is a rearrangement, so that the rearranged series diverges against determined.

Under the rearrangement is defined as a bijective mapping of the set of natural numbers to itself ( a permutation ).

Grounds

One divides the sequence into two subsequences and that only the non - negative and the negative sequence elements of contain. For example:

The rows and both are definitely divergent. In fact, if one of the two series is convergent, then the other would also be convergent, since it is the difference of the original series and the first series could ( with inserted zeros ) write. But that would also be absolutely convergent, in contradiction to the assumption.

In particular, the fact that there are infinitely many terms with positive sign and infinite number of terms with negative sign follows.

Construction of the rearrangement

A series that converges to the real number that can be constructed as follows: You summed as long as positive followers to, for the first time until you exceed the target ( in case this is the empty sum ).

Subsequently, as long as a negative result then summed members until the partial sum is less than the value.

Afterwards, go alternately proceed with non -negative and negative sequence elements. This consideration results in a rearrangement of the original series, there is a null sequence because of their convergence. Figuratively, one can imagine this process as:

In every small - strip around now are the followers of Partialreihe for sufficiently large indices. The so- rearranged series converges to.

Is such to select the non- th Partialreihe negative sequence elements in the above structure so that the number is exceeded. Then select the smallest index, unused, negative follower. The resulting rearrangement diverges to. The case can be treated accordingly.

Example

The example of the alternating harmonic series, the effect of reordering is to be shown. This series is convergent but not absolutely convergent: The series

Converges, while the harmonic series

Diverges. Although the alternating harmonic series in the normal representation converges to ln ( 2), it can be rearranged to converge to any other number, or even diverges according to the Riemann rearrangement theorem. In the example, it is achieved only by reordering the limit value ln ( 2) / 2.

The usual spelling of this series is:

If the reordered summands are obtained:

In general, this sum is made up of three blocks:

Such a block can be transformed to:

The total sum is thus exactly half of the alternating harmonic series:

Steinitzscher rearrangement theorem

The steinitzscher rearrangement theorem is a generalization of Riemannian Umordnungssatzes. Is a convergent series, then the amount of the threshold values ​​of all the convergent rearranged rows

An affine subspace of. In particular, in the complex plane is then either a point, a line or completely. The series is absolutely convergent if and only if only a single point contains.

Swell

  • Harro Heuser: Textbook of analysis. Part I, Teubner -Verlag.
  • Sequences and series
  • Set ( mathematics)
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