# 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 $\sum_{n=1}^{\infty}a_{n}$ converges but $\sum_{n=1}^{\infty}|a_{n}|=\infty$, i.e, $\sum a_{n}$ is conditionally convergent. Let $-\infty\leq\alpha<\beta\leq\infty$ be arbitrary. Then there exists a rearrangement $(a_{n^{\prime}})$ such that

 $\liminf_{N}\sum_{n^{\prime}=1}^{N}a_{n^{\prime}}=\alpha\ \ \ \mbox{and}\ \ \ % \limsup_{N}\sum_{n^{\prime}=1}^{N}a_{n^{\prime}}=\beta.$

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

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 $a_{n}\neq 0$. 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

 $\displaystyle a^{+}_{1}$ $\displaystyle+a^{+}_{2}+\cdots+a^{+}_{m_{1}}$ $\displaystyle-a^{-}_{1}-a^{-}_{2}-\cdots-a^{-}_{k_{2}}$ $\displaystyle+a^{+}_{m_{1}+1}+a^{+}_{2}+\cdots+a^{+}_{m_{2}}$ $\displaystyle-a^{-}_{k_{1}+1}-a^{-}_{2}-\cdots-a^{-}_{k_{2}}+$ $\displaystyle\ \ \ \vdots$

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 $\alpha_{j}\rightarrow\alpha$ , $\beta_{j}\rightarrow\beta$, $\alpha_{n}\leq\beta_{n}$ and $\beta_{1}>0$. We choose $m_{1}$ such that $\sum_{n=1}^{m_{1}}\geq\beta_{1}$ but $\sum_{n=1}^{m_{1}-1}<\beta_{1}$. We choose $k_{1}$ such that $\sum_{n=1}^{m_{1}}-\sum_{n=1}^{k_{1}}\leq\alpha_{1}$ 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 $\alpha$. $\ \ \ \ \ \Box$

## References

Title Riemann’s theorem on rearrangements RiemannsTheoremOnRearrangements 2013-03-22 17:31:52 2013-03-22 17:31:52 Gorkem (3644) Gorkem (3644) 23 Gorkem (3644) Theorem msc 40A05 Riemann series theorem UnconditionallyConvergent FiniteChangesInConvergentSeries FiniteChangesInConvergentSeries2