# Thread: Rearrangement of conditionally convergent series

1. ## Rearrangement of conditionally convergent series

Suppose that $\displaystyle \sum^\infty_{n=1} u_n = s,$ where series converge conditionally and $\displaystyle s'>s.$ Help me please in proving following assertion: there exists permutation $\displaystyle \sigma:\mathbb{N}\rightarrow \mathbb{N}$ with properties:
1) $\displaystyle u_n\geq 0\implies \sigma(n)=n;$
2) $\displaystyle \sum^\infty_{n=1} u_{\sigma(n)}=s'.$

I can prove that for any sequences of positive numbers, $\displaystyle (b_n), (B_n)$ such that $\displaystyle b_n\rightarrow 0, \ \sum^\infty_{n=1} b_n=\infty, \ 0<B_1<B_2<\ldots<B_n\rightarrow \infty$, there exists a permutation $\displaystyle \sigma:\mathbb{N}\rightarrow \mathbb{N}$, for which
$\displaystyle \forall k\geq 1 \sum^k_{n=1} b_{\sigma(n)}\leq B_k.$
But I can't apply this properly to initial problem

2. Originally Posted by Georgii
Suppose that $\displaystyle \sum^\infty_{n=1} u_n = s,$ where series converge conditionally and $\displaystyle s'>s.$ Help me please in proving following assertion: there exists permutation $\displaystyle \sigma:\mathbb{N}\rightarrow \mathbb{N}$ with properties:
1) $\displaystyle u_n\geq 0\implies \sigma(n)=n;$
2) $\displaystyle \sum^\infty_{n=1} u_{\sigma(n)}=s'.$

I can prove that for any sequences of positive numbers, $\displaystyle (b_n), (B_n)$ such that $\displaystyle b_n\rightarrow 0, \ \sum^\infty_{n=1} b_n=\infty, \ 0<B_1<B_2<\ldots<B_n\rightarrow \infty$, there exists a permutation $\displaystyle \sigma:\mathbb{N}\rightarrow \mathbb{N}$, for which
$\displaystyle \forall k\geq 1 \sum^k_{n=1} b_{\sigma(n)}\leq B_k.$
But I can't apply this properly to initial problem

Read the proof of Riemann's Theorem in series.

Tonio

3. My question deals nothing with the proof of Riemann's theorem, because the permutation which is built there doesn't satisfy property 1) of mentioned statement. And I don't know whether it is possible to build a permutation with properties 1) and 2) using the method of proof of Riemann's theorem.