Riemann series theorem
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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.
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[edit] Definitions
A series converges if there exists a value such that the sequence of the partial sums
converges to . That is, for any ε > 0, there exists an integer N such that if , then
- .
A series converges conditionally if the series converges but the series 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 , then .
[edit] Statement of the theorem
Suppose that
is a sequence of real numbers, and that is conditionally convergent. Let M be a real number. Then there exists a permutation σ(n) of the sequence such that
There also exists a permutation σ(n) such that
The sum can also be rearranged to diverge to or to fail to approach any limit, finite or infinite.
[edit] Examples
The alternating harmonic series is a classic example of a conditionally convergent series:
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,
and rearrange the terms:
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:
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
this series can in fact be written:
which is half the usual sum.
[edit] References
- Weisstein, Eric (2005). Riemann Series Theorem. Retrieved May 16, 2005.