Riemann series theorem

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.

Come esempio, 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, eccetera.) to give a series that converges to a different sum: 1 + 1/2 − 1 + 1/3 + 1/4 − 1/2 + ⋯ = ln 2. Più generalmente, 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.

Contenuti 1 Definizioni 2 Enunciato del teorema 3 Alternating harmonic series 3.1 Changing the sum 3.2 Getting an arbitrary sum 4 Prova 4.1 Existence of a rearrangement that sums to any positive real M 4.2 Existence of a rearrangement that diverges to infinity 4.3 Existence of a rearrangement that fails to approach any limit, finito o infinito 5 generalizzazioni 5.1 Sierpiński theorem 5.2 Il teorema di Steinitz 6 Guarda anche 7 References Definitions A series {textstyle sum _{n=1}^{infty }un_{n}} converges if there exists a value {stile di visualizzazione ell } such that the sequence of the partial sums {stile di visualizzazione (S_{1},S_{2},S_{3},ldot ),quad S_{n}=somma _{k=1}^{n}un_{K},} converge a {stile di visualizzazione ell } . Questo è, for any ε > 0, there exists an integer N such that if n ≥ N, poi {displaystyle leftvert S_{n}-ell rightvert leq epsilon .} A series converges conditionally if the series {textstyle sum _{n=1}^{infty }un_{n}} converges but the series {textstyle sum _{n=1}^{infty }leftvert a_{n}rightvert } diverges.

A permutation is simply a bijection from the set of positive integers to itself. This means that if {displaystyle sigma } is a permutation, then for any positive integer {stile di visualizzazione b,} there exists exactly one positive integer {stile di visualizzazione a} tale che {displaystyle sigma (un)=b.} In particolare, Se {displaystyle xneq y} , poi {displaystyle sigma (X)neq sigma (y)} .

Statement of the theorem Suppose that {stile di visualizzazione (un_{1},un_{2},un_{3},ldot )} is a sequence of real numbers, e quello {textstyle sum _{n=1}^{infty }un_{n}} is conditionally convergent. Permettere {stile di visualizzazione M} be a real number. Then there exists a permutation {displaystyle sigma } tale che {somma dello stile di visualizzazione _{n=1}^{infty }un_{sigma (n)}=M.} There also exists a permutation {displaystyle sigma } tale che {somma dello stile di visualizzazione _{n=1}^{infty }un_{sigma (n)}=infty .} The sum can also be rearranged to diverge to {displaystyle -infty } or to fail to approach any limit, finito o infinito.

Alternating harmonic series Changing the sum The alternating harmonic series is a classic example of a conditionally convergent series: {somma dello stile di visualizzazione _{n=1}^{infty }{frac {(-1)^{n+1}}{n}}} is convergent, whereas {somma dello stile di visualizzazione _{n=1}^{infty }sinistra|{frac {(-1)^{n+1}}{n}}Giusto|=somma _{n=1}^{infty }{frac {1}{n}}} is the ordinary harmonic series, which diverges. Although in standard presentation the alternating harmonic series converges to ln(2), 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, {stile di visualizzazione 1-{frac {1}{2}}+{frac {1}{3}}-{frac {1}{4}}+cdot } and rearrange the terms: {stile di visualizzazione 1-{frac {1}{2}}-{frac {1}{4}}+{frac {1}{3}}-{frac {1}{6}}-{frac {1}{8}}+{frac {1}{5}}-{frac {1}{10}}-{frac {1}{12}}+cdot } 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 generale, the sum is composed of blocks of three: {stile di visualizzazione {frac {1}{2k-1}}-{frac {1}{2(2k-1)}}-{frac {1}{4K}},quad k=1,2,dots .} 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). Da {stile di visualizzazione {frac {1}{2k-1}}-{frac {1}{2(2k-1)}}={frac {1}{2(2k-1)}},} this series can in fact be written: {stile di visualizzazione {inizio{allineato}&{frac {1}{2}}-{frac {1}{4}}+{frac {1}{6}}-{frac {1}{8}}+{frac {1}{10}}+cdot +{frac {1}{2(2k-1)}}-{frac {1}{2(2K)}}+cdots \={}&{frac {1}{2}}sinistra(1-{frac {1}{2}}+{frac {1}{3}}-cdot a destra)={frac {1}{2}}ln(2)fine{allineato}}} 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 {stile di visualizzazione 1+{1 Sopra 2}+{1 Sopra 3}+cdot +{1 over n}=gamma +ln n+o(1),} where γ is the Euler–Mascheroni constant, and where the notation o(1) denotes a quantity that depends upon the current variable (qui, 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 {stile di visualizzazione {1 Sopra 2}+{1 Sopra 4}+{1 Sopra 6}+cdot +{1 over 2q}={1 Sopra 2},gamma +{1 Sopra 2}ln q+o(1),} and by taking the difference, one sees that the sum of p odd terms satisfies {stile di visualizzazione {1}+{1 Sopra 3}+{1 Sopra 5}+cdot +{1 over 2p-1}={1 Sopra 2},gamma +{1 Sopra 2}ln p+ln 2+o(1).} 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): {stile di visualizzazione {1}+{1 Sopra 3}+cdot +{1 over 2a-1}-{1 Sopra 2}-{1 Sopra 4}-cdot -{1 over 2b}+{1 over 2a+1}+cdot +{1 over 4a-1}-{1 over 2b+2}-cdot } 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, quindi {stile di visualizzazione S_{(a+b)n}={1 Sopra 2}ln p+ln 2-{1 Sopra 2}ln q+o(1)={1 Sopra 2}ln(a/b)+ln 2+o(1).} It follows that the sum of this rearranged series is {stile di visualizzazione {1 Sopra 2}ln(a/b)+ln 2=ln left(2{mq {a/b}}Giusto).} Suppose now that, più generalmente, 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. Quindi, the sum of such a rearrangement will be {stile di visualizzazione ln sinistra(2{mq {r}}Giusto),} 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.

Proof Existence of a rearrangement that sums to any positive real M For simplicity, this proof assumes first that an ≠ 0 for every n. The general case requires a simple modification, indicato di seguito. Recall that a conditionally convergent series of real terms has both infinitely many negative terms and infinitely many positive terms. Primo, define two quantities, {stile di visualizzazione a_{n}^{+}} e {stile di visualizzazione a_{n}^{-}} di: {stile di visualizzazione a_{n}^{+}={frac {un_{n}+|un_{n}|}{2}},quad a_{n}^{-}={frac {un_{n}-|un_{n}|}{2}}.} Questo è, the series {textstyle sum _{n=1}^{infty }un_{n}^{+}} includes all an positive, with all negative terms replaced by zeroes, and the series {textstyle sum _{n=1}^{infty }un_{n}^{-}} includes all an negative, with all positive terms replaced by zeroes. Da {textstyle sum _{n=1}^{infty }un_{n}} 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 {stile di visualizzazione a_{n}^{+}} so that their sum exceeds M. Suppose we require p terms – then the following statement is true: {somma dello stile di visualizzazione _{n=1}^{p-1}un_{n}^{+}leq M 0 because the partial sums of {stile di visualizzazione a_{n}^{+}} tend to {displaystyle +infty } . Discarding the zero terms one may write {somma dello stile di visualizzazione _{n=1}^{p}un_{n}^{+}=a_{sigma (1)}+cdot +a_{sigma (m_{1})},quad a_{sigma (j)}>0, sigma (1) 0), or as image of m1 + 1 (if a1 < 0). Now repeat the process of adding just enough positive terms to exceed M, starting with n = p + 1, and then adding just enough negative terms to be less than M, starting with n = q + 1. Extend σ in an injective manner, in order to cover all terms selected so far, and observe that a2 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 {textstyle sum {a_{sigma (n)}}} . After the first change of direction, each partial sum of {textstyle sum {a_{sigma (n)}}} differs from M by at most the absolute value {displaystyle a_{p_{j}}^{+}} or {displaystyle |a_{q_{j}}^{-}|} of the term that appeared at the latest change of direction. But {textstyle sum {a_{n}}} converges, so as n tends to infinity, each of an, {displaystyle a_{p_{j}}^{+}} and {displaystyle a_{q_{j}}^{-}} go to 0. Thus, the partial sums of {textstyle sum {a_{sigma (n)}}} tend to M, so the following is true: {displaystyle sum _{n=1}^{infty }a_{sigma (n)}=M.} 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 Ak of integers and a real number Sk are defined. For every k > 0, the induction defines the value {textstyle sigma (K)} , the set Ak consists of the values {textstyle sigma (j)} for j ≤ k and Sk is the partial sum of the rearranged series. The definition is as follows: Per k = 0, the induction starts with A0 empty and S0 = 0. For every k ≥ 0, there are two cases: if Sk ≤ M, poi {textstyle sigma (k+1)} is the smallest integer n ≥ 1 such that n is not in Ak and an ≥ 0; if Sk > M, poi {textstyle sigma (k+1)} is the smallest integer n ≥ 1 such that n is not in Ak and an < 0. In both cases one sets {displaystyle A_{k+1}=A_{k}cup {sigma (k+1)},;quad S_{k+1}=S_{k}+a_{sigma (k+1)}.} 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 {textstyle sum _{i=1}^{infty }a_{i}} be a conditionally convergent series. The following is a proof that there exists a rearrangement of this series that tends to {displaystyle infty } (a similar argument can be used to show that {displaystyle -infty } can also be attained). Let {displaystyle p_{1}be the sequence of indexes such that each {stile di visualizzazione a_{p_{io}}} is positive, e definire {stile di visualizzazione n_{1}to be the indexes such that each {stile di visualizzazione a_{n_{io}}} is negative (again assuming that {stile di visualizzazione a_{io}} is never 0). Each natural number will appear in exactly one of the sequences {stile di visualizzazione (p_{io})} e {stile di visualizzazione (n_{io}).} Permettere {stile di visualizzazione b_{1}} be the smallest natural number such that {somma dello stile di visualizzazione _{io=1}^{b_{1}}un_{p_{io}}geq |un_{n_{1}}|+1.} Such a value must exist since {stile di visualizzazione (un_{p_{io}}),} the subsequence of positive terms of {stile di visualizzazione (un_{io}),} diverges. Allo stesso modo, permettere {stile di visualizzazione b_{2}} be the smallest natural number such that: {somma dello stile di visualizzazione _{i=b_{1}+1}^{b_{2}}un_{p_{io}}geq |un_{n_{2}}|+1,} e così via. This leads to the permutation {stile di visualizzazione (sigma (1),sigma (2),sigma (3),ldot )=(p_{1},p_{2},ldot ,p_{b_{1}},n_{1},p_{b_{1}+1},p_{b_{1}+2},ldot ,p_{b_{2}},n_{2},ldot ).} And the rearranged series, {textstyle sum _{io=1}^{infty }un_{sigma (io)},} then diverges to {displaystyle infty } . From the way the {stile di visualizzazione b_{io}} were chosen, it follows that the sum of the first {stile di visualizzazione b_{1}+1} terms of the rearranged series is at least 1 and that no partial sum in this group is less than 0. Allo stesso modo, the sum of the next {stile di visualizzazione b_{2}-b_{1}+1} terms is also at least 1, and no partial sum in this group is less than 0 o. Continuing, this suffices to prove that this rearranged sum does indeed tend to {displaystyle infty .} Existence of a rearrangement that fails to approach any limit, finite or infinite In fact, Se {textstyle sum _{n=1}^{infty }un_{n}} is conditionally convergent, then there is a rearrangement of it such that the partial sums of the rearranged series form a dense subset of {displaystyle mathbb {R} .} [citazione necessaria] Generalizations Sierpiński theorem In Riemann's theorem, the permutation used for rearranging a conditionally convergent series to obtain a given value in {displaystyle mathbb {R} tazza {infty ,-infty }} may have arbitrarily many non-fixed points, cioè. 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.[1][2][3] This question has also been explored using the notion of ideals: per esempio, Wilczyński proved that it is sufficient to consider rearrangements whose set of non-fixed indices belongs to the ideal of asymptotic density zero (questo è, it is sufficient to rearrange a set of indices of asymptotic density zero).[4] Filipów and Szuca proved that other ideals also have this property.[5] Steinitz's theorem Main article: Lévy–Steinitz theorem Given a converging series {textstyle sum {un_{n}}} of complex numbers, several cases can occur when considering the set of possible sums for all series {textstyle sum {un_{sigma (n)}}} obtained by rearranging (permuting) the terms of that series: the series {textstyle sum {un_{n}}} may converge unconditionally; poi, all rearranged series converge, and have the same sum: the set of sums of the rearranged series reduces to one point; the series {textstyle sum {un_{n}}} may fail to converge unconditionally; if S denotes the set of sums of those rearranged series that converge, poi, either the set S is a line L in the complex plane C, della forma {stile di visualizzazione L={a+tb:latta matematicabb {R} },quad a,sono matematicabb {C} , bneq 0,} or the set S is the whole complex plane C. Più generalmente, 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. See also Absolute convergence § Rearrangements and unconditional convergence References ^ Sierpiński, Waclaw (1910). "Contribution à la théorie des séries divergentes". Comptes rendus de l'Académie des Sciences Varsovie. 3: 89–93. ^ Sierpinski, Waclaw (1910). "Remarque sur la théorème de Riemann relatif aux séries semi-convergentes". Prac. Stuoia. Fiz. XXI: 17–20. ^ Sierpinski, Waclaw (1911). "Sur une propriété des séries qui ne sont pas absolument convergentes". Bulletin International de l'Académie des Sciences de Cracovie, Séries A. 149–158. ^ Wilczyński, Władysław (2007). "On Riemann derangement theorem". Słup. pr. Mat.-Fiz. 4: 79–82. ^ Filipów, Rafał; Szuca, Piotr (febbraio 2010). "Rearrangement of conditionally convergent series on a small set". Journal of Mathematical Analysis and Applications. 362 (1): 64–71. doi:10.1016/j.jmaa.2009.07.029. Apostolo, Tom (1975). Calcolo, Volume 1: One-variable Calculus, with an Introduction to Linear Algebra. Banaszczyk, Wojciech (1991). "Capitolo 3.10 The Lévy–Steinitz theorem". Additive subgroups of topological vector spaces. Appunti delle lezioni in matematica. vol. 1466. Berlino: Springer-Verlag. pp. 93–109. ISBN 3-540-53917-4. SIG 1119302. Kadets, V. M.; Kadets, M. io. (1991). "Capitolo 1.1 The Riemann theorem, Capitolo 6 The Steinitz theorem and B-convexity". Rearrangements of series in Banach spaces. Traduzioni di monografie matematiche. vol. 86 (Translated by Harold H. McFaden from the Russian-language (Tartu) 1988 ed.). Provvidenza, RI: Società matematica americana. pp. iv+123. ISBN 0-8218-4546-2. SIG 1108619. Kadets, Mikhail I.; Kadets, Vladimir M. (1997). "Capitolo 1.1 The Riemann theorem, Capitolo 2.1 Steinitz's theorem on the sum range of a series, Capitolo 7 The Steinitz theorem and B-convexity". Series in Banach spaces: Conditional and unconditional convergence. Teoria degli operatori: Anticipi e applicazioni. vol. 94. Translated by Andrei Iacob from the Russian-language. Basilea: Birkhauser Verlag. pp. viii+156. ISBN 3-7643-5401-1. SIG 1442255. Weisstein, Eric (2005). Riemann Series Theorem. 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