Riemann series theorem
In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges. This implies that a series of real numbers is absolutely convergent if and only if it is unconditionally convergent. As an example, the series 1 − 1 + 1/2 − 1/2 + 1/3 − 1/3 + ⋯ 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 1 + 1 + 1/2 + 1/2 + 1/3 + 1/3 + ⋯, which sums to infinity. Thus the original series is conditionally convergent, and can be rearranged (by taking the first two positive terms followed by the first negative term, followed by the next two positive terms and then the next negative term, etc.) to give a series that converges to a different sum: 1 + 1/2 − 1 + 1/3 + 1/4 − 1/2 + ⋯ =  ln 2. More generally, using this procedure with "p" positives followed by "q" negatives gives the sum ln("p"/"q"). Other rearrangements give other finite sums or do not converge to any sum. Definitions. A series formula_1 converges if there exists a value formula_2 such that the sequence of the partial sums converges to formula_2. That is, for any "ε" > 0, there exists an integer "N" such that if "n" ≥ "N", then A series converges conditionally if the series formula_1 converges but the series formula_7 diverges. A permutation is simply a bijection from the set of positive integers to itself . This means that if formula_8 is a permutation, then for any positive integer formula_9 there exists exactly one positive integer formula_10 such that formula_11 In particular, if formula_12, then formula_13. Statement of the theorem. Suppose that formula_14 is a sequence of real numbers, and that formula_15 is conditionally convergent. Let formula_16 be a real number. Then there exists a permutation formula_8 such that There also exists a permutation formula_8 such that The sum can also be rearranged to diverge to formula_21 or to fail to approach any limit, finite or infinite. Alternating harmonic series. Changing the sum. The alternating harmonic  series is a classic example of a conditionally convergent series: formula_22 is convergent, whereas formula_23 is the ordinary harmonic series, which diverges. Although in standard presentation the alternating harmonic series converges to , its terms can be arranged to converge to any number, or even to diverge. One instance of this is as follows. Begin with the series written in the usual order, and rearrange the terms: where the pattern is: the first two terms are 1 and −1/2, whose sum is 1/2. The next term is −1/4. The next two terms are 1/3 and −1/6, whose sum is 1/6. The  next term is −1/8. The next two terms are 1/5 and −1/10, whose sum is 1/10. In general, the sum is composed of blocks of three: This is indeed a rearrangement of the alternating harmonic series: every odd integer occurs once positively, and the even integers occur once each, negatively (half of them as multiples of 4, the other half as twice odd integers). Since this series can in fact be written: which is half the usual sum. Getting an arbitrary sum. An efficient way to recover and generalize the result of the previous section is to use the fact  that where "γ" is the Euler–Mascheroni constant, and where the notation "o"(1) denotes a quantity that depends upon the current variable (here, the variable is "n") in such a way that this quantity goes to 0 when the variable tends to infinity. 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 , 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 positive odd terms and negative even terms, hence It follows that the sum of this rearranged series is Suppose now that, more generally, a rearranged series of the alternating harmonic series is organized in such a way that the ratio 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 . Proof. Existence of a rearrangement that sums to any positive real "M". For simplicity, this proof assumes first that "a" ≠ 0 for every "n". The general case requires a simple modification, given below. Recall that a conditionally convergent series of real terms has both infinitely many negative terms and infinitely  many positive terms. First, define two quantities, formula_36 and formula_37 by: That is, the series formula_39 includes all "a" positive, with all negative terms replaced by zeroes, and the series formula_40 includes all "a" negative, with all positive terms replaced by zeroes. Since formula_1 is conditionally convergent, both the positive and the negative series diverge. Let "M" be a positive real number. Take, in order, just enough positive terms formula_42 so that their sum exceeds "M". Suppose we require "p" terms – then the following statement is true: This is possible for any "M" > 0 because the partial sums of  formula_42 tend to formula_45. Discarding the zero terms one may write Now we add just enough negative terms formula_47, say "q" of them, so that the resulting sum is less than "M". This is always possible because the partial sums of formula_47 tend to formula_21. Now we have: Again, one may write with The map "σ" is injective, and 1 belongs to the range of "σ", either as image of 1 (if "a" > 0), or as image of (if "a" < 0). Now repeat the process of adding just enough positive terms to exceed "M", starting with , and then  adding just enough negative terms to be less than "M", starting with . Extend "σ" in an injective manner, in order to cover all terms selected so far, and observe that must have been selected now or before, thus 2 belongs to the range of this extension. The process will have infinitely many such ""changes of direction"". One eventually obtains a rearrangement  . After the first change of direction, each partial sum of  differs from "M" by at most the absolute value formula_53 or formula_54 of the term that appeared at the latest change of direction. But converges, so as "n" tends  to infinity, each of "a", formula_53 and formula_56 go to 0. Thus, the partial sums of  tend to "M", so the following is true: The same method can be used to show convergence to "M" negative or zero. One can now give a formal inductive definition of the rearrangement "σ", that works in general. For every integer "k" ≥ 0, a finite set "A" of integers and a real number "S" are defined. For every "k" > 0, the induction defines the value "σ"("k"), the set "A" consists of the values "σ"("j") for "j" ≤ "k" and "S" is the  partial sum of the rearranged series. The definition is as follows: It can be proved, using the reasonings above, that "σ" is a permutation of the integers and that the permuted series converges to the given real number "M". Existence of a rearrangement that diverges to infinity. Let formula_59 be a conditionally convergent series. The following is a proof that there exists a rearrangement of this series that tends to formula_60 (a similar argument can be used to show that formula_21 can also be attained). Let formula_62 be the sequence of indexes such that each formula_63 is positive, and define  formula_64 to be the indexes such that each formula_65 is negative (again assuming that formula_66 is never 0). Each natural number will appear in exactly one of the sequences formula_67 and formula_68 Let formula_69 be the smallest natural number such that Such a value must exist since formula_71 the subsequence of positive terms of formula_72 diverges. Similarly, let formula_73 be the smallest natural number such that: and so on. This leads to the permutation And the rearranged series, formula_76 then diverges to formula_60. From the way the formula_78 were chosen, it follows that the sum of the first formula_79 terms  of the rearranged series is at least 1 and that no partial sum in this group is less than 0. Likewise, the sum of the next formula_80 terms is also at least 1, and no partial sum in this group is less than 0 either. Continuing, this suffices to prove that this rearranged sum does indeed tend to formula_81 Existence of a rearrangement that fails to approach any limit, finite or infinite. In fact, if formula_82 is conditionally convergent, then there is a rearrangement of it such that the partial sums of the rearranged series form a dense subset of  formula_83 Generalizations. Sierpiński theorem. In Riemann's theorem, the permutation used for rearranging a conditionally convergent series to obtain a given value in formula_84 may have arbitrarily many non-fixed points, i.e. all the indexes of the terms of the series may be rearranged. One may ask if it is possible to rearrange only the indexes in a smaller set so that a conditionally convergent series converges to an arbitrarily chosen real number or diverges to (positive or negative) infinity. The answer of this question is yes but only to smaller values: 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. This question has also been explored using the notion of ideals: for instance, Wilczyński proved that it is sufficient to consider rearrangements whose set of non-fixed indices belongs to the ideal of asymptotic density zero (that is, it is sufficient to rearrange a set of indices of asymptotic density zero). Filipów and Szuca proved that other ideals also have this property. Steinitz's theorem. Given a converging series of  complex numbers, several cases can occur when considering the set of possible sums for all series obtained by rearranging (permuting) the terms of that series: More generally, given a converging series of vectors in a finite-dimensional real vector space "E", the set of sums of converging rearranged series is an affine subspace of "E".