Riemann series theorem

In mathematics, the Riemann series theorem (also called the Riemann rearrangement 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.

Definitions

A series $sum\_\{n=1\}^infty\; a\_n$ converges if there exists a value $ell$ such that the sequence of the partial sums

:$left\; \{\; S\_1,\; S\_2,\; S\_3,\; dots\; ight\; \}$

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

:$left\; |\; S\_n\; -\; ell\; ight\; vert\; le\; epsilon$.

A series converges conditionally if the series $sum\_\{n=1\}^infty\; a\_n$ converges but the series $sum\_\{n=1\}^infty\; left\; |\; a\_n\; ight\; vert$ diverges.

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

tatement of the theorem

Suppose that

:$left\; \{\; a\_1,\; a\_2,\; a\_3,\; dots\; ight\; \}$

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 $sigma\; (n)$ of the sequence such that

:$sum\_\{n=1\}^infty\; a\_\{sigma\; (n)\}\; =\; M.$

There also exists a permutation $sigma\; (n)$ such that

:$sum\_\{n=1\}^infty\; a\_\{sigma\; (n)\}\; =\; infty.$

The sum can also be rearranged to diverge to $-infty$ or to fail to approach any limit, finite or infinite.

Examples

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

:$sum\_\{n=1\}^infty\; frac\{(-1)^\{n+1\{n\}$

is convergent, while

:

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. One instance of this is as follows. Begin with the series written in the usual order,

:$1\; -\; frac\{1\}\{2\}\; +\; frac\{1\}\{3\}\; -\; frac\{1\}\{4\}\; +\; cdots$

and rearrange the terms:

:$1\; -\; frac\{1\}\{2\}\; -\; frac\{1\}\{4\}\; +\; frac\{1\}\{3\}\; -\; frac\{1\}\{6\}\; -\; frac\{1\}\{8\}\; +\; frac\{1\}\{5\}\; -\; frac\{1\}\{10\}\; +\; cdots$

where the pattern is: the first two terms are 1 and −1/2, whose sum is 1/2. The next term is −1/4. The next two terms are 1/3 and −1/6, whose sum is 1/6. The next term is −1/8. The next two terms are 1/5 and −1/10, whose sum is 1/10. In general, the sum is composed of blocks of three:

:$frac\{1\}\{2k\; -\; 1\}\; -\; frac\{1\}\{2(2k\; -\; 1)\}\; -\; frac\{1\}\{4k\},quad\; k\; =\; 1,\; 2,\; dots.$

This is indeed a rearrangement of the alternating harmonic series: every odd integer occurs once positively, and the even integers occur once each, negatively (half of them as multiples of 4, the other half as twice odd integers). Since

:$frac\{1\}\{2k\; -\; 1\}\; -\; frac\{1\}\{2(2k\; -\; 1)\}\; =\; frac\{1\}\{2(2k\; -\; 1)\},$

this series can in fact be written:

:$frac\{1\}\{2\}\; -\; frac\{1\}\{4\}\; +\; frac\{1\}\{6\}\; -\; frac\{1\}\{8\}\; +\; frac\{1\}\{10\}\; +\; cdots\; +\; frac\{1\}\{2(2k\; -\; 1)\}\; -\; frac\{1\}\{2(2k)\}\; +\; cdots$

:$=\; frac\{1\}\{2\}left(1\; -\; frac\{1\}\{2\}\; +\; frac\{1\}\{3\}\; +\; cdots\; ight)\; =\; frac\{1\}\{2\}\; ln(2)$which is half the usual sum.

References

*Weisstein, Eric (2005). [http://mathworld.wolfram.com/RiemannSeriesTheorem.html Riemann Series Theorem] . Retrieved May 16, 2005.