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| 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]] is [[conditionally convergent]], then its terms can be arranged in a [[permutation]] so that the new series converges to any given value, or [[divergent series|diverges]].
| | THE BIG APPLE - Jay Victorino was standing outside his mother's residence when he was grabbed by police, and he says if she hadn't come downstairs to determine him he would've been arrested on a trespassing cost.<br><br>As soon as IRAS is notified of the property switch, all property tax related correspondence together with property tax bills can be sent to you as the new proprietor. Because the property owner, you'll be chargeable for all property tax on the property. Hence, it is important that you just, by way of your lawyer, enquire if there are any excellent or future property tax liabilities, and make provisions for the vendor to reimburse you for the property tax amount that's to be borne by him.<br><br>CPF financial savings can only be used for properties constructed on freehold or leasehold land with a remaining lease of a minimum of 30 years provided the remaining lease can final you as much as at the very least 80 years outdated. This is a computation of your whole month-to-month debt obligations (e.g. existing house loan, new house loan that you are making use of for, automobile loan, overdraft facilities, different credit score services and recurrent debt obligations) to your total month-to-month income. Month-to-month repayment instalments for all property loans and other debt obligations are not to exceed a TDSR of 60%. That is to encourage prudent borrowing by households. Flying by means of the Asian skies, Invoice Barnett involves the unsettling conclusion that the property industry is being overrun by the cult of made-up terms<br><br>b. A property held by 2 or extra persons as tenants in common means that every owner has a specified or distinct share or proportion within the property. For instance, such share might be on an equal foundation (50%-50%) or 90% - 10% foundation etc. Every proprietor may be able to dispose his share in the property both by sale or underneath his Will. Upon the death of one proprietor, the surviving co-owner(s) do(es) not take over the deceased's curiosity in the property as is in the case of joint tenancy. His share shall be passed underneath his will or in accordance with the legislation of intestacy, as the case could also be.<br><br>D25) Admiralty / Woodlands HDB Common Room D19) Hougang / Punggol / Sengkang HDB House for Hire D27) Sembawang / Yishun HDB Grasp Room D12) Balestier / Toa Payo HDB Frequent Room D18) Pasir Ris / Tampines HDB Dwelling for Hire D28) Seletar / Yio Chu Kang HDB Home for Hire D27) Sembawang / Yishun HDB Residence for Hire D25) Admiralty / Woodlands HDB Residence for Rent D13) Potong Pasir / Machpherson HDB House for Rent D12) Balestier / Toa Payo HDB Dwelling for Hire Seminar Leader Mr Josh [http://anyzblog.q--q.net/?document_srl=979981 anyzblog.q--q.net] Ng is a full-time lecturer in Ngee Ann Polytechnic's College of Design & Surroundings and has been training and instructing in the space of actual estate for several years. Stamp responsibility on buy (Please check with the IRAS website for data on stamp responsibility) Great Funding Worth! HDB Scheme Any DBSS BTO<br><br>Lease Home District 25 Rent Home District 26 Rent Home District 27 Hire Home District 28 Southeast Asia Property Assessment - MarketPulse Q1 2014 Softening in residential property costs continue, led by 2.eight per cent decline in the index for Rest of Central Area Thakral to take a position up to A$46.2 mil in Brisbane property project Different miscellaneous cost Housing loan Down-cost Supply of funding HDB flat with mortgage from HDB 10% of purchase worth or market valuation, whichever is lower CPF financial savings and cash (if CPF savings are insufficient) (A) Consumers taking out a loan from a financial institution and with no different excellent housing mortgage(s), whether or not on HDB flat or non-public property. Make sure you purchase a house you could afford in the long run. Residential Units Totally Offered. view this challenge<br><br>For properties owned by multiple owner, all homeowners are collectively answerable for paying property tax. For correspondence purpose, we will often tackle to the owner who's listed first within the Discover of Transfer filed by the vendor's lawyer. Hence, it will be significant that you simply as the new owner or one of the new house owners, inform the seller's lawyer the one that will be answerable for the property tax matters. Once the transfer record is updated, we might be corresponding with the person on all property tax issues, including cost of property tax. |
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| ==Definitions==
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| A series <math>\sum_{n=1}^\infty a_n</math> [[convergent series|converges]] if there exists a value <math>\ell</math> such that the [[sequence]] of the partial sums
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| :<math>\left \{ S_1, \ S_2, \ S_3, \dots \right \}, \quad S_n = \sum_{k=1}^n a_k,</math>
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| converges to <math>\ell</math>. That is, for any ''ε'' > 0, there exists an integer ''N'' such that if ''n'' ≥ ''N'', then
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| :<math>\left\vert S_n - \ell \right\vert \le \ \epsilon.</math>
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| A series [[conditional convergence|converges conditionally]] if the series <math>\sum_{n=1}^\infty a_n</math> converges but the series <math>\sum_{n=1}^\infty \left\vert a_n \right\vert</math> diverges.
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| A permutation is simply a [[bijection]] from the [[Set (mathematics)|set]] of [[positive integer]]s to itself. This means that if <math>\sigma</math> is a permutation, then for any positive integer <math>b</math>, there exists exactly one positive integer <math>a</math> such that <math>\sigma (a) = b</math>. In particular, if <math>x \ne y</math>, then <math>\sigma (x) \ne \sigma (y)</math>.
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| ==Statement of the theorem==
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| Suppose that
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| :<math>\left \{ a_1, \ a_2, \ a_3, \dots \right \}</math>
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| is a sequence of [[real number]]s, and that <math>\sum_{n=1}^\infty a_n</math> is conditionally convergent. Let <math>M</math> be a real number. Then there exists a [[permutation]] <math>\sigma (n)</math> of the sequence such that
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| :<math>\sum_{n=1}^\infty a_{\sigma (n)} = M.</math>
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| There also exists a permutation <math>\sigma (n)</math> such that
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| :<math>\sum_{n=1}^\infty a_{\sigma (n)} = \infty.</math>
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| The sum can also be rearranged to diverge to <math>-\infty</math> or to fail to approach any limit, finite or infinite.
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| ==Examples==
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| ===Changing the sum===
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| The [[alternating harmonic series]] is a classic example of a conditionally convergent series:
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| :<math>\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}</math>
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| is convergent, while
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| :<math>\sum_{n=1}^\infty \bigg| \frac{(-1)^{n+1}}{n} \bigg|</math>
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| is the ordinary [[harmonic series (mathematics)|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,
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| :<math>1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots</math>
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| and rearrange the terms:
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| :<math>1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - \frac{1}{10} + \cdots</math>
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| 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:
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| :<math>\frac{1}{2k - 1} - \frac{1}{2(2k - 1)} - \frac{1}{4k},\quad k = 1, 2, \dots.</math>
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| 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
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| :<math>\frac{1}{2k - 1} - \frac{1}{2(2k - 1)} = \frac{1}{2(2k - 1)},</math>
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| this series can in fact be written:
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| :<math>\frac{1}{2} - \frac{1}{4} + \frac{1}{6} - \frac{1}{8} + \frac{1}{10} + \cdots + \frac{1}{2(2k - 1)} - \frac{1}{2(2k)} + \cdots</math>
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| :<math>= \frac{1}{2}\left(1 - \frac{1}{2} + \frac{1}{3} + \cdots\right) = \frac{1}{2} \ln(2)</math>
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| which is half the usual sum.
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| ===Getting an arbitrary sum===
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| An efficient way to recover and generalize the result of the previous section is to use the fact that
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| :<math>1 + {1 \over 2} + {1 \over 3} + \cdots + {1 \over n} = \gamma + \ln n + o(1),</math>
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| where ''γ'' is the [[Euler–Mascheroni constant]], and where the [[Big O notation|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.
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| It follows that the sum of ''q'' even terms satisfies
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| :<math>{1 \over 2} + {1 \over 4} + {1 \over 6} + \cdots + {1 \over 2 q} = {1 \over 2} \, \gamma + {1 \over 2} \ln q + o(1),</math>
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| and by taking the difference, one sees that the sum of ''p'' odd terms satisfies
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| :<math>{1} + {1 \over 3} + {1 \over 5} + \cdots + {1 \over 2 p - 1} = {1 \over 2} \, \gamma + {1 \over 2} \ln p + \ln 2 + o(1).</math>
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| 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 {{nowrap|''a'' {{=}} ''b'' {{=}} 1}}, the example in the preceding section corresponds to ''a'' = 1, ''b'' = 2):
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| :<math>{1} + {1 \over 3} + \cdots + {1 \over 2 a - 1} - {1 \over 2} - {1 \over 4} - \cdots - {1 \over 2 b} + {1 \over 2 a + 1} + \cdots + {1 \over 4 a - 1} - {1 \over 2b + 2} - \cdots</math>
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| Then the partial sum of order (''a''+''b'')''n'' of this rearranged series contains {{nowrap|''p'' {{=}} ''a'' ''n''}} positive odd terms and {{nowrap|''q'' {{=}} ''b'' ''n''}} negative even terms, hence
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| :<math>S_{(a+b)n} = {1 \over 2} \ln p + \ln 2 - {1 \over 2} \ln q + o(1) = {1 \over 2} \ln(a/b) + \ln 2 + o(1).</math>
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| It follows that the sum of this rearranged series is
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| :<math>{1 \over 2} \ln(a/b) + \ln 2 = \ln\bigl( 2 \sqrt{a/b} \bigr).</math>
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| Suppose now that, more generally, a rearranged series of the alternating harmonic series is organized in such a way that the ratio {{nowrap|''p''<sub>''n''</sub> / ''q''<sub>''n''</sub>}} 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
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| :<math>\ln\bigl( 2 \sqrt{r} \bigr),</math>
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| 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 {{nowrap|to  e<sup>2''x''</sup> /  4}}.
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| ==Proof==
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| For simplicity, this proof assumes first that ''a''<sub>''n''</sub> ≠ 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, <math>a_n^+</math> and <math>a_n^-</math> by:
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| :<math>a_{n}^{+} = \frac{a_n + |a_n|}{2}, \quad a_{n}^{-} = \frac{a_n - |a_n|}{2}.</math>
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| That is, the series <math>\sum_{n=1}^\infty a_n^{+}</math> includes all ''a''<sub>''n''</sub> positive, with all negative terms replaced by zeroes, and the series <math>\sum_{n=1}^\infty a_n^{-}</math> includes all ''a''<sub>''n''</sub> negative, with all positive terms replaced by zeroes. Since <math>\sum_{n=1}^\infty a_n</math> 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 <math>a_{n}^{+}</math> so that their sum exceeds ''M''. Suppose we require ''p'' terms – then the following statement is true:
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| :<math>\sum_{n=1}^{p-1} a_{n}^{+} \leq M < \sum_{n=1}^{p} a_{n}^{+}.</math>
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| This is possible for any ''M'' > 0 because the partial sums of <math>a_{n}^{+}</math> tend to <math>+\infty</math>. Discarding the zero terms one may write
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| :<math>\sum_{n=1}^{p} a_{n}^{+} = a_{\sigma(1)} + \cdots + a_{\sigma(m_1)}, \quad a_{\sigma(j)} > 0, \ \ \sigma(1) < \ldots < \sigma(m_1) = p.</math>
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| Now we add just enough negative terms <math>a_{n}^{-}</math>, say ''q'' of them, so that the resulting sum is less than ''M''. This is always possible because the partial sums of <math>a_{n}^{-}</math> tend to <math>-\infty</math>. Now we have:
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| :<math>\sum_{n=1}^{p} a_{n}^{+} + \sum_{n=1}^{q} a_{n}^{-} < M \leq \sum_{n=1}^{p} a_{n}^{+} + \sum_{n=1}^{q - 1} a_{n}^{-}.</math>
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| Again, one may write
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| :<math>\sum_{n=1}^{p} a_{n}^{+} + \sum_{n=1}^{q} a_{n}^{-} = a_{\sigma(1)} + \cdots + a_{\sigma(m_1)} + a_{\sigma(m_1+1)} + \cdots + a_{\sigma(n_1)},</math>
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| with
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| :<math> \sigma(m_1+1) < \ldots < \sigma(n_1) = q.</math>
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| Note that ''σ'' is injective, and that 1 belongs to the range of ''σ'', either as image of 1 (if ''a''<sub>1</sub> > 0), or as image of {{nowrap|''m'' <sub>1</sub> + 1}} (if ''a''<sub>1</sub> < 0). Now repeat the process of adding just enough positive terms to exceed ''M'', starting with {{nowrap|''n'' {{=}} ''p'' + 1}}, and then adding just enough negative terms to be less than ''M'', starting with {{nowrap|''n'' {{=}} ''q'' + 1}}. Extend ''σ'' in an injective manner, in order to cover all terms selected so far, and observe that {{nowrap|''a'' <sub>2</sub>}} 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 {{nowrap|∑ ''a''<sub>''σ'' (''n'')</sub>}}. After the first change of direction, each partial sum of {{nowrap|∑ ''a''<sub>''σ'' (''n'')</sub>}} differs from ''M'' by at most the absolute value <math>a_{p_j}^{+}</math> or <math>|a_{q_j}^{-}|</math> of the term that appeared at the latest change of direction. But {{nowrap|∑ ''a''<sub>''n''</sub>}} converges, so as ''n'' tends to infinity, each of ''a''<sub>''n''</sub>, <math>a_{p_j}^{+}</math> and <math>a_{q_j}^{-}</math> go to 0. Thus, the partial sums of {{nowrap|∑ ''a''<sub>''σ'' (''n'')</sub>}} tend to ''M'', so the following is true:
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| :<math>\sum_{n=1}^\infty a_{\sigma(n)} = M.</math>
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| The same method can be used to show convergence to ''M'' negative or zero.
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| 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''<sub>''k''</sub> of integers and a real number ''S''<sub>''k''</sub> are defined. For every ''k'' > 0, the induction defines the value ''σ''(''k''), the set ''A''<sub>''k''</sub> consists of the values ''σ''(''j'') for ''j'' ≤ ''k'' and ''S''<sub>''k''</sub> is the partial sum of the rearranged series. The definition is as follows:
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| * For ''k'' = 0, the induction starts with ''A''<sub>0</sub> empty and ''S''<sub>0</sub> = 0.
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| * For every ''k'' ≥ 0, there are two cases: if ''S''<sub>''k''</sub> ≤ ''M'', then ''σ''(''k''+1) is the smallest integer ''n'' ≥ 1 such that ''n'' is not in ''A''<sub>''k''</sub> and ''a''<sub>''n''</sub> ≥ 0; if ''S''<sub>''k''</sub> > ''M'', then ''σ''(''k''+1) is the smallest integer ''n'' ≥ 1 such that ''n'' is not in ''A''<sub>''k''</sub> and ''a''<sub>''n''</sub> < 0. In both cases one sets
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| :<math>A_{k+1} = A_k \cup \{\sigma(k+1)\} \, ; \quad S_{k+1} = S_k + a_{\sigma(k+1)}.</math>
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| 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''.
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| == Generalization ==
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| {{see also|Steinitz's theorem}}
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| Given a converging series {{nowrap|∑ ''a''<sub>''n''</sub>}} of [[complex number]]s, several cases can occur when considering the set of possible sums for all series {{nowrap|∑ ''a''<sub>''σ'' (''n'')</sub>}} obtained by rearranging (permuting) the terms of that series:
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| * the series {{nowrap|∑ ''a''<sub>''n''</sub>}} may converge unconditionally; then, all rearranged series converge, and have the same sum: the set of sums of the rearranged series reduces to one point;
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| * the series {{nowrap|∑ ''a''<sub>''n''</sub>}} may fail to converge unconditionally; if ''S'' denotes the set of sums of those rearranged series that converge, then, either the set ''S'' is a line ''L'' in the complex plane '''C''', of the form
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| ::<math>L = \{a + t b : t \in \mathbf{R} \}, \quad a, b \in \mathbf{C}, \ b \ne 0,</math>
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| :or the set ''S'' is the whole complex plane '''C'''.
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| 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 space|affine subspace]] of ''E''.
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| ==References==
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| *Apostol, Tom (1975). ''Calculus, Volume 1: One-variable Calculus, with an Introduction to Linear Algebra.''
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| *{{cite book |last=Banaszczyk |first=Wojciech |chapter=Chapter 3.10 The Lévy–Steinitz theorem |title=Additive subgroups of topological vector spaces |series=Lecture Notes in Mathematics |volume=1466 |publisher=Springer-Verlag |location=Berlin |year=1991 |pages=93–109 |isbn=3-540-53917-4 |mr=1119302}}
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| *{{cite book |last1=Kadets |first1=V. M. |last2=Kadets |first2=M. I. |authorlink2=Mikhail Kadets |chapter=Chapter 1.1 The Riemann theorem, Chapter 6 The Steinitz theorem and ''B''-convexity |title=Rearrangements of series in Banach spaces |edition=Translated by Harold H. McFaden from the Russian-language (Tartu) 1988 |series=Translations of Mathematical Monographs |volume=86 |publisher=American Mathematical Society |location=Providence, RI |year=1991 |pages=iv+123 |isbn=0-8218-4546-2 |mr=1108619}}
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| *{{cite book |last1=Kadets |first1=Mikhail I. |last2=Kadets |first2=Vladimir M. |chapter=Chapter 1.1 The Riemann theorem, Chapter 2.1 Steinitz's theorem on the sum range of a series, Chapter 7 The Steinitz theorem and ''B''-convexity |title=Series in Banach spaces: Conditional and unconditional convergence |others=Translated by Andrei Iacob from the Russian-language |series=Operator Theory: Advances and Applications |volume=94 |publisher=Birkhäuser Verlag |location=Basel |year=1997 |pages=viii+156 |isbn=3-7643-5401-1 |mr=1442255}}
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| *Weisstein, Eric (2005). [http://mathworld.wolfram.com/RiemannSeriesTheorem.html Riemann Series Theorem]. Retrieved May 16, 2005.
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| [[Category:Mathematical series]]
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| [[Category:Theorems in analysis]]
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| [[Category:Permutations]]
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| [[Category:Summability theory]]
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THE BIG APPLE - Jay Victorino was standing outside his mother's residence when he was grabbed by police, and he says if she hadn't come downstairs to determine him he would've been arrested on a trespassing cost.
As soon as IRAS is notified of the property switch, all property tax related correspondence together with property tax bills can be sent to you as the new proprietor. Because the property owner, you'll be chargeable for all property tax on the property. Hence, it is important that you just, by way of your lawyer, enquire if there are any excellent or future property tax liabilities, and make provisions for the vendor to reimburse you for the property tax amount that's to be borne by him.
CPF financial savings can only be used for properties constructed on freehold or leasehold land with a remaining lease of a minimum of 30 years provided the remaining lease can final you as much as at the very least 80 years outdated. This is a computation of your whole month-to-month debt obligations (e.g. existing house loan, new house loan that you are making use of for, automobile loan, overdraft facilities, different credit score services and recurrent debt obligations) to your total month-to-month income. Month-to-month repayment instalments for all property loans and other debt obligations are not to exceed a TDSR of 60%. That is to encourage prudent borrowing by households. Flying by means of the Asian skies, Invoice Barnett involves the unsettling conclusion that the property industry is being overrun by the cult of made-up terms
b. A property held by 2 or extra persons as tenants in common means that every owner has a specified or distinct share or proportion within the property. For instance, such share might be on an equal foundation (50%-50%) or 90% - 10% foundation etc. Every proprietor may be able to dispose his share in the property both by sale or underneath his Will. Upon the death of one proprietor, the surviving co-owner(s) do(es) not take over the deceased's curiosity in the property as is in the case of joint tenancy. His share shall be passed underneath his will or in accordance with the legislation of intestacy, as the case could also be.
D25) Admiralty / Woodlands HDB Common Room D19) Hougang / Punggol / Sengkang HDB House for Hire D27) Sembawang / Yishun HDB Grasp Room D12) Balestier / Toa Payo HDB Frequent Room D18) Pasir Ris / Tampines HDB Dwelling for Hire D28) Seletar / Yio Chu Kang HDB Home for Hire D27) Sembawang / Yishun HDB Residence for Hire D25) Admiralty / Woodlands HDB Residence for Rent D13) Potong Pasir / Machpherson HDB House for Rent D12) Balestier / Toa Payo HDB Dwelling for Hire Seminar Leader Mr Josh anyzblog.q--q.net Ng is a full-time lecturer in Ngee Ann Polytechnic's College of Design & Surroundings and has been training and instructing in the space of actual estate for several years. Stamp responsibility on buy (Please check with the IRAS website for data on stamp responsibility) Great Funding Worth! HDB Scheme Any DBSS BTO
Lease Home District 25 Rent Home District 26 Rent Home District 27 Hire Home District 28 Southeast Asia Property Assessment - MarketPulse Q1 2014 Softening in residential property costs continue, led by 2.eight per cent decline in the index for Rest of Central Area Thakral to take a position up to A$46.2 mil in Brisbane property project Different miscellaneous cost Housing loan Down-cost Supply of funding HDB flat with mortgage from HDB 10% of purchase worth or market valuation, whichever is lower CPF financial savings and cash (if CPF savings are insufficient) (A) Consumers taking out a loan from a financial institution and with no different excellent housing mortgage(s), whether or not on HDB flat or non-public property. Make sure you purchase a house you could afford in the long run. Residential Units Totally Offered. view this challenge
For properties owned by multiple owner, all homeowners are collectively answerable for paying property tax. For correspondence purpose, we will often tackle to the owner who's listed first within the Discover of Transfer filed by the vendor's lawyer. Hence, it will be significant that you simply as the new owner or one of the new house owners, inform the seller's lawyer the one that will be answerable for the property tax matters. Once the transfer record is updated, we might be corresponding with the person on all property tax issues, including cost of property tax.