[1][2] As an example, the series converges to 0 (for a sufficiently large number of terms, the partial sum gets arbitrarily near to 0); but replacing all terms with their absolute values gives which sums to infinity.
to give a series that converges to a different sum, such as which evaluates to ln 2.
More generally, using this procedure with p positives followed by q negatives gives the sum ln(p/q).
It is a basic result that the sum of finitely many numbers does not depend on the order in which they are added.
The observation that the sum of an infinite sequence of numbers can depend on the ordering of the summands is commonly attributed to Augustin-Louis Cauchy in 1833.
[3] He analyzed the alternating harmonic series, showing that certain rearrangements of its summands result in different limits.
Around the same time, Peter Gustav Lejeune Dirichlet highlighted that such phenomena are ruled out in the context of absolute convergence, and gave further examples of Cauchy's phenomenon for some other series which fail to be absolutely convergent.
[6] Riemann's theorem is now considered as a basic part of the field of mathematical analysis.
[7] For any series, one may consider the set of all possible sums, corresponding to all possible rearrangements of the summands.
Riemann’s theorem can be formulated as saying that, for a series of real numbers, this set is either empty, a single point (in the case of absolute convergence), or the entire real number line (in the case of conditional convergence).
In this formulation, Riemann’s theorem was extended by Paul Lévy and Ernst Steinitz to series whose summands are complex numbers or, even more generally, elements of a finite-dimensional real vector space.
[8][9] They proved that the set of possible sums forms a real affine subspace.
Extensions of the Lévy–Steinitz theorem to series in infinite-dimensional spaces have been considered by a number of authors.
This series can be demonstrated to be greater than zero by the proof of Leibniz's theorem using that the second partial sum is half.
Hence, the value of the sequence is shown to depend on the order in which series is computed.
It follows that the sum of q even terms satisfies and by taking the difference, one sees that the sum of p odd terms satisfies Suppose that two positive integers a and b are given, and that a rearrangement of the alternating harmonic series is formed by taking, in order, a positive terms from the alternating harmonic series, followed by b negative terms, and repeating this pattern at infinity (the alternating series itself corresponds to a = b = 1, the example in the preceding section corresponds to a = 1, b = 2): Then the partial sum of order (a + b)n of this rearranged series contains p = an positive odd terms and q = bn negative even terms, hence It follows that the sum of this rearranged series is[12] Suppose now that, more generally, a rearranged series of the alternating harmonic series is organized in such a way that the ratio pn/qn between the number of positive and negative terms in the partial sum of order n tends to a positive limit r. Then, the sum of such a rearrangement will be and this explains that any real number x can be obtained as sum of a rearranged series of the alternating harmonic series: it suffices to form a rearrangement for which the limit r is equal to e2x/ 4.
Riemann's description of the theorem and its proof reads in full:[13] … infinite series fall into two distinct classes, depending on whether or not they remain convergent when all the terms are made positive.
Clearly now an arbitrarily given value C can be obtained by a suitable reordering of the terms.
Now, since the number a as well as the numbers b become infinitely small with increasing index, so also are the deviations from C. If we proceed sufficiently far in the series, the deviation becomes arbitrarily small, that is, the series converges to C.This can be given more detail as follows.
includes all an positive, with all negative terms replaced by zeroes, and the series
Now let q1 be the smallest positive integer such that This number exists because the partial sums of
The result may be viewed as a new sequence Furthermore the partial sums of this new sequence converge to M. This can be seen from the fact that for any i, with the first inequality holding due to the fact that pi+1 has been defined as the smallest number larger than pi which makes the second inequality true; as a consequence, it holds that Since the right-hand side converges to zero due to the assumption of conditional convergence, this shows that the (pi+1 + qi)'th partial sum of the new sequence converges to M as i increases.
The above proof of Riemann's original formulation only needs to be modified so that pi+1 is selected as the smallest integer larger than pi such that and with qi+1 selected as the smallest integer larger than qi such that The choice of i+1 on the left-hand sides is immaterial, as it could be replaced by any sequence increasing to infinity.
The above proof only needs to be modified so that pi+1 is selected as the smallest integer larger than pi such that and with qi+1 selected as the smallest integer larger than qi such that This directly shows that the sequence of partial sums contains infinitely many entries which are larger than 1, and also infinitely many entries which are less than −1, so that the sequence of partial sums cannot converge.
, and study the real numbers that the series can sum to if we are only allowed to permute indices in
With this notation, we have: Sierpiński proved that rearranging only the positive terms one can obtain a series converging to any prescribed value less than or equal to the sum of the original series, but larger values in general can not be attained.
(that is, it is sufficient to rearrange a set of indices of asymptotic density zero).
of complex numbers, several cases can occur when considering the set of possible sums for all series
"Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält".
Translated by Baker, Roger; Christenson, Charles; Orde, Henry.