Riemann series theorem

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In mathematics, the Riemann series theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series is conditionally convergent, then its terms can be arranged in a permutation so that the series converges to any given value, or even diverges.

Contents

[edit] Definitions

A series \sum_{n=1}^\infty a_n converges if there exists a value \ell such that the sequence of the partial sums

\left \{ S_1, \ S_2, \ S_3, ... \right \}

converges to \ell. That is, for any ε > 0, there exists an integer N such that if n \ge \ N, then

\left | S_n - \ell \right \vert \le \ \epsilon.

A series converges conditionally if the series \sum_{n=1}^\infty a_n converges but the series \sum_{n=1}^\infty \left | a_n \right \vert diverges.

A permutation is simply a bijection from the set of positive integers to itself. This means that if σ(n) is a permutation, then for any positive integer b, there exists a positive integer a such that σ(a) = b. Furthermore, if x \ne y, then \sigma (x) \ne \sigma (y).

[edit] Consequences of the theorem

Suppose that

\left \{ a_1, \ a_2, \ a_3, ... \right \}

is a sequence of real numbers, and that \sum_{n=1}^\infty a_n is conditionally convergent. Let M be a real number. Then there exists a permutation σ(n) of the sequence such that

\sum_{n=1}^\infty a_{\sigma (n)} = M.

There also exists a permutation σ(n) such that

\sum_{n=1}^\infty a_{\sigma (n)} = \infty.

[edit] Examples

The alternating harmonic series is a classic example of a conditionally convergent function:

\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}

is convergent, while

\sum_{n=1}^\infty \bigg| \frac{(-1)^{n+1}}{n} \bigg|

is the ordinary harmonic series, which diverges. Although in standard presentation, the alternating harmonic series converges to ln(2), its terms can be arranged to converge to any number, or even to diverge.

[edit] Reference

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