15

I was looking through my lecture notes and got puzzled by the following fact: if we want to find the value of some infinite series we are allowed to rearrange only the finite number of its terms. To visualize this consider the alternating harmonic series: $$\sum_{n=1}^\infty(-1)^{k-1}\frac1k=1-\frac12+\frac13-\frac14+\frac15-+\dots=0.693147...$$

But if we rearrange the terms as follows the value of the series gets influenced by this action: $$1+\frac13-\frac12+\frac15+\frac17-\frac14+\dots=1.03972...$$

So commutativity of addition isn't true on infinity? How was it obtained and how can it be proved?

Dmitry Kazakov
  • 882
  • Because you have an alternating series and the limit is the limit common to the summation over odd and even terms and you truncated and you did not use the same terms. – Claude Leibovici Aug 02 '14 at 07:13
  • 3
    You are right about commutativity not valid in an infinite sum. This is because the accepted meaning of a convergent series is based on the partial sums. The sequence of partial sums will differ if you rearrange the numbers. In fact, it is a theorem that you can get ANY number by rearranging the series that converges conditionally. You can even make it divergent. The situation is different if your series converges absolutely, in which case rearranging would not affect the infinite sum. – Tunococ Aug 02 '14 at 07:40
  • 1
    The higher-dimensional version of this phenomenon is addressed at http://mathoverflow.net/questions/29333/can-a-conditionally-convergent-series-of-vectors-be-rearranged-to-give-any-limit – KCd Aug 02 '14 at 09:23

3 Answers3

6

We have the strong intuition that changing the order of the terms in a sum never changes the sum because it never does for a finite sum. We didn't just accept without proof that it never does for a finite sum. Rather, it's a theorem that can be proven from the associativity and commutativity of addition despite the fact that the associative law only states associativity for a sum of 3 terms and the commutative law only states commutativity for a sum of 2 terms. That proof can't be generalized to the case of an infinite sum. Note that the positive terms in the first series add to infinity and the negative terms in it add to negative infinity. The reason the 2 series have different sums is because as n gets larger, the sum of the negative terms within the first n terms of the second series never gets anywhere near keeping up with the sum of the negative terms within the first n terms of the first series.

Timothy
  • 793
5

It's quite easy to think up elementary counter-examples.

For example, consider the series

$$1-1+1-1+1-1+...=(1-1)+(1-1)+(1-1)+...\\ =0+0+0+...\\ =0.$$

If it is permissible to commute an infinite number of terms, you can rearrange the series into,

$$1-1+1-1+1-1+1-...=1+(-1+1)+(-1+1)+(-1+1)\\ =1+0+0+0+...\\ =1,$$

implying $0=1$. Generally speaking, $0=1$ is undesirable result.

David H
  • 29,921
2

Why does the order of summation of the terms of an infinite series influence its value ?

Because, although the series is convergent, it is not absolutely convergent.

Lucian
  • 48,334
  • 2
  • 83
  • 154
  • So if the series is absolutely convergent we can apply the commutativity of addition? – Dmitry Kazakov Aug 02 '14 at 08:22
  • If the series is absolutely convergent, then the order of summation is irrelevant, and it does not affect the final outcome in any way. – Lucian Aug 02 '14 at 08:24
  • 11
    This does not answer the question: Why and where does noncommutativity sneak in? – Christian Blatter Aug 02 '14 at 08:40
  • 1
    Absolute convergence not only justifies that series rearrangements don't affect the sum, but this condition exactly captures that property: any series of real numbers that is not absolutely convergent can be rearranged to converge to anything at all (i.e., the order of the terms in the series generally matters). This is what makes Fourier series generally so subtle, since such series can be convergent but not absolutely convergent. That absolutely convergent series are those that can be rearranged without changing the sum is one reason the absolute convergence hypothesis appears a lot. – KCd Aug 02 '14 at 09:27
  • @ChristianBlatter so nice to see you on Math StackExchange. Right now reading your Analysis I/II Skript for engineers :) – Dmitry Kazakov Aug 02 '14 at 09:33
  • @KCd can you give an example of Fourier series which could be rearranged so as to sum to a radically different function? – Ruslan Apr 02 '17 at 16:32
  • @Ruslan this theorem on rearrangements is (1) a property of series of numbers, so for a Fourier series it is a separate theorem for each value of $x$, and (2) usually not a constructive result: even though the theorem says you can rearrange the terms of a conditionally convergent series to have any desired value (including $\pm \infty$), it is not usually practical to make such a rearrangement explicit. If you want to see explicit examples, since I see you are from СПБ I suggest you take a look at the examples in section 388 of volume II in Фихтенгольц. The proof of the theorem is there too! – KCd Apr 02 '17 at 21:52
  • @DmitryKazakov: Could you please un-accept my answer, so that I may delete it ? Thank you. – Lucian Jun 09 '22 at 15:27