# Conditionally convergent series

I am looking for the original reference discussing a specific, elementary example of a rearrangement of series converging to a value different from the original series. In what follows, I give some (meager) context and make explicit the example whose origin I am trying to locate.

Apparently, Dirichlet was first to note that a conditionally convergent series may be rearranged so that the resulting series converges to a different value. This was in 1827, during his work on the convergence of Fourier series. He also proved, 10 years later, that rearranging an absolutely convergent series has no effect on its limit.

In 1853, Riemann proved his rearrangement theorem, although it was not published until 1866, as part of his Habilitationsschrift on representation of functions as trigonometric series. See here for Riemann's papers.

Theorem (Riemann). Given any α≤β in the extended reals, a conditionally convergent series of reals can be rearranged so that the liminf of the partial sums of the rearranged series is α, while the limsup is β. In particular, any real can be obtained as the sum of some rearrangement of the original series.

Prior to Riemann's theorem, Ohm had obtained interesting related result, that were later extended by Schlömilch, and Pringsheim, see here and here for statements of some of their results, and modern extensions. They are usually discussed in terms of rearrangements of the alternating harmonic series, although their results are more general.

The alternating harmonic series is the series $$1 - 1/2 + 1/3 - 1/4 + 1/5 - ... = \ln(2)$$

Ohm's theorem predicts the value of the series obtained if we fix positive integers $p$ and $q$, and the terms of the series are rearranged so that first we add the first$p$ positive terms, then the first $q$ negative terms, then the next p positive terms, then the next q negative ones, etc. For instance, if $p=1$ and $q=2$, the rearrangement is $$1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - \cdots$$

Ohm's theorem states that this series converges to $$\ln(2) + (1/2) \ln(p/q)$$ In particular, for $p=1$, $q=2$, the limit is $(1/2) \ln(2)$.

Nowadays, when discussing rearrangements (say, in calculus textbooks), it is common to find this specific example, evaluated as follows: $$1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - \cdots = (1 - 1/2) - 1/4 + (1/3 - 1/6) - 1/8 + (1/5 - 1/10) - 1/12 + \cdots = (1/2)(1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + \cdots)$$

The proof of the general result is more intricate, but introductory discussions of rearrangements omit it, while the argument above is completely basic and intuitive. (The other example that tends to be found in introductory discussions is $$1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + \cdots = (3/2) \ln(2)$$

corresponding to $p=2$, $q=1$ in Ohm's theorem. This example can be verified by a similar, only slightly more involved trick, see here.)

I am trying to locate the first place where this specific proof is mentioned. Manning's classical (1906) book Irrational numbers and their representation by sequences and series mentions the example, in pages 97, 98, but the only source it mentions is "(Laurent)", without further bibliographical details.

Finally, although perhaps off topic for this list, if a similar elementary observation can be used to prove Ohm's theorem, or a large fragment of it, I would be grateful for details. The general case is not difficult, but the proofs I am aware of are not quite as suitable for discussion in an introductory course.

• What exactly do you mean by "this specific proof": a proof of Ohm's theorem? a calculation of a special case of Ohm's theorem (the one with $p = 1$ and $q = 2$, or with $p = 2$ and $q = 1$, or any choice of $p$ and $q$ besides both equal to $1$)? Please be clearer about exactly what you are seeking. – KCd Sep 3 '18 at 9:39
• Thank you for your interest. I meant the specific evaluation for $p=1$ and $q=2$ that I show in the displayed text following the "evaluated as follows:". In any case, more generally, I am also interested in early references discussing rearrangements; I have been told some of the earliest references are essentially unknown and are not mentioned in discussions of the history of these results. – Andrés E. Caicedo Sep 3 '18 at 14:22
• The earliest reference I know about of the awareness that rearranging a convergent series can lead to a convergent series with a different value was by Dirichlet on the 5th page of his paper proving his theorem on primes in arithmetic progression (link to article is in the reference section of the Wikipedia page on that theorem), and I can see from your blog post andrescaicedo.wordpress.com/2014/11/16/…, created soon after writing your question above, that you knew about this. What is Dirichlet's 1827 paper? – KCd Sep 3 '18 at 15:49