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

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