# Riemann’s theorem on rearrangements

If the map $n\mapsto {n}^{\prime}$ is a bijection on $\mathbb{N}$, we say that the sequence $({a}_{{n}^{\prime}})$ is a rearrangement of $({a}_{n})$.

The following theorem, which is due to Riemann, shows that the convergence of a conditionally convergent series depends so much on the order of its terms; in particular, a conditionally convergent series can be made to converge to any real number by changing the order of its terms.

Theorem (Riemann series theorem^{}).
Let $({a}_{n})$ be a sequence in $\mathbb{R}$ such that ${\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}$ converges but ${\sum}_{n=1}^{\mathrm{\infty}}|{a}_{n}|=\mathrm{\infty}$, i.e, $\sum {a}_{n}$ is conditionally convergent. Let $$ be arbitrary. Then there exists a rearrangement $({a}_{{n}^{\prime}})$ such that

$$\underset{N}{lim\; inf}\sum _{{n}^{\prime}=1}^{N}{a}_{{n}^{\prime}}=\alpha \mathit{\hspace{1em}\hspace{1em}\u2006}\text{and}\mathit{\hspace{1em}\hspace{1em}\u2006}\underset{N}{lim\; sup}\sum _{{n}^{\prime}=1}^{N}{a}_{{n}^{\prime}}=\beta .$$ |

Proof. Let ${a}_{n}^{+}=\mathrm{max}\{0,{a}_{n}\}$ and ${a}_{n}^{-}=\mathrm{min}\{0,-{a}_{n}\}$. Then we have ${a}_{n}={a}_{n}^{+}-{a}_{n}^{-}$ and $|{a}_{n}|={a}_{n}^{+}+{a}_{n}^{-}$. Since $$, both $\sum {a}_{n}^{+}$ and $\sum {a}_{n}^{-}$ diverge or converge simultaneously. But since $\sum |{a}_{n}|=\mathrm{\infty}$, we see that at least one of $\sum {a}_{n}^{+}$ and $\sum {a}_{n}^{-}$ must diverge. It follows that $\sum {a}_{n}^{+}=\mathrm{\infty}$ and $\sum {a}_{n}^{-}=\mathrm{\infty}$.

Also by the $n$th term test, ${lim}_{n}{a}_{n}^{+}={lim}_{n}{a}_{n}^{-}=0$.

Now we pass to subsequence of $({a}_{n}^{+})$ by removing all terms with ${a}_{n}^{+}=0$ and ${a}_{n}\ne 0$. For ${a}_{n}^{-}$, we remove all terms with ${a}_{n}^{-}=0$. Let us denote the subsequences still as $({a}_{n}^{+})$ and $({a}_{n}^{-})$. Since only zeros have been removed, $\sum {a}_{n}^{+}$ and $\sum {a}_{n}^{-}$ are still divergent.

Now we will define integers ${m}_{j}$ and ${k}_{j}$ for $j\in \mathbb{N}$, and consider the series

${a}_{1}^{+}$ | $+{a}_{2}^{+}+\mathrm{\cdots}+{a}_{{m}_{1}}^{+}$ | ||

$-{a}_{1}^{-}-{a}_{2}^{-}-\mathrm{\cdots}-{a}_{{k}_{2}}^{-}$ | |||

$+{a}_{{m}_{1}+1}^{+}+{a}_{2}^{+}+\mathrm{\cdots}+{a}_{{m}_{2}}^{+}$ | |||

$-{a}_{{k}_{1}+1}^{-}-{a}_{2}^{-}-\mathrm{\cdots}-{a}_{{k}_{2}}^{-}+$ | |||

$\mathrm{}\mathit{\hspace{1em}}\mathrm{\vdots}$ |

This series is clearly a rearrangement of $\sum {a}_{n}$, by our choice of the subsequences ${a}_{n}^{+}$ and ${a}_{n}^{-}$.

We pick up two sequences ${\alpha}_{j}$ and ${\beta}_{j}$ such that ${\alpha}_{j}\to \alpha $ , ${\beta}_{j}\to \beta $, ${\alpha}_{n}\le {\beta}_{n}$ and ${\beta}_{1}>0$. We choose ${m}_{1}$ such that ${\sum}_{n=1}^{{m}_{1}}\ge {\beta}_{1}$ but $$. We choose ${k}_{1}$ such that ${\sum}_{n=1}^{{m}_{1}}-{\sum}_{n=1}^{{k}_{1}}\le {\alpha}_{1}$ but ${\sum}_{n=1}^{{m}_{1}}-{\sum}_{n=1}^{{k}_{1}-1}>{\alpha}_{1}$. We continue this way, inductively.

Since ${lim}_{n}{a}_{n}^{+}={lim}_{n}{a}_{n}^{-}=0$, the subsequences of the sequence of partial sums that end with ${a}_{{m}_{j}}^{+}$ and ${a}_{{k}_{j}}^{-}$ converge to $\beta $ and $\alpha $, and it can be seen that no subsequence can be found with a limit larger than $\beta $ or lower than $\alpha $. $\mathrm{\square}$

## References

- 1 Rudin, W., Principles of Mathematical Analysis, McGraw Hill, 1976.

Title | Riemann’s theorem on rearrangements |
---|---|

Canonical name | RiemannsTheoremOnRearrangements |

Date of creation | 2013-03-22 17:31:52 |

Last modified on | 2013-03-22 17:31:52 |

Owner | Gorkem (3644) |

Last modified by | Gorkem (3644) |

Numerical id | 23 |

Author | Gorkem (3644) |

Entry type | Theorem |

Classification | msc 40A05 |

Synonym | Riemann series theorem |

Related topic | UnconditionallyConvergent |

Related topic | FiniteChangesInConvergentSeries |

Related topic | FiniteChangesInConvergentSeries2 |