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 of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges. This implies that a series of real numbers is absolutely convergent if and only if it is unconditionally convergent.

As an example, the series

1

−

1

+

1

2

−

1

2

+

1

3

−

1

3

+

1

4

−

1

4

+

…

{\displaystyle 1-1+{\frac {1}{2}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{3}}+{\frac {1}{4}}-{\frac {1}{4}}+\dots }

converges to 0 (for a sufficiently large number of terms, the partial sum gets arbitrarily near to 0); but replacing all terms with their absolute values gives

1

+

1

+

1

2

+

1

2

+

1

3

+

1

3

+

…

{\displaystyle 1+1+{\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{3}}+\dots }

which sums to infinity. Thus, the original series is conditionally convergent, and can be rearranged (by taking the first two positive terms followed by the first negative term, followed by the next two positive terms and then the next negative term, etc.) to give a series that converges to a different sum, such as

1

+

1

2

−

1

+

1

3

+

1

4

−

1

2

+

…

{\displaystyle 1+{\frac {1}{2}}-1+{\frac {1}{3}}+{\frac {1}{4}}-{\frac {1}{2}}+\dots }

which evaluates to ln 2. More generally, using this procedure with p positives followed by q negatives gives the sum ln(p/q). Other rearrangements give other finite sums or do not converge to any sum.

History

It is a basic result that the sum of finitely many numbers does not depend on the order in which they are added. For example, 2 + 6 + 7 = 7 + 2 + 6. The observation that the sum of an infinite sequence of numbers can depend on the ordering of the summands is commonly attributed to Augustin-Louis Cauchy in 1833. He analyzed the alternating harmonic series, showing that certain rearrangements of its summands result in different limits.