Riemann’s theorem on rearrangements
If the map is a bijection on , we say that the sequence is a rearrangement of .
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 be a sequence in such that converges but , i.e, is conditionally convergent. Let be arbitrary. Then there exists a rearrangement such that
Proof. Let and . Then we have and . Since , both and diverge or converge simultaneously. But since , we see that at least one of and must diverge. It follows that and .
Also by the th term test, .
Now we pass to subsequence of by removing all terms with and . For , we remove all terms with . Let us denote the subsequences still as and . Since only zeros have been removed, and are still divergent.
Now we will define integers and for , and consider the series
This series is clearly a rearrangement of , by our choice of the subsequences and .
We pick up two sequences and such that , , and . We choose such that but . We choose such that but . We continue this way, inductively.
Since , the subsequences of the sequence of partial sums that end with and converge to and , and it can be seen that no subsequence can be found with a limit larger than or lower than .
- 1 Rudin, W., Principles of Mathematical Analysis, McGraw Hill, 1976.
|Title||Riemann’s theorem on rearrangements|
|Date of creation||2013-03-22 17:31:52|
|Last modified on||2013-03-22 17:31:52|
|Last modified by||Gorkem (3644)|
|Synonym||Riemann series theorem|