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| In [[mathematics]], a '''divergent series''' is an [[infinite series]] that is not [[Convergent series|convergent]], meaning that the infinite [[sequence]] of the [[partial sum]]s of the series does not have a finite [[limit of a sequence|limit]].
| | Neil Bush, the brother of former U.S. President George W. Bush, is using Singapore as a springboard to bankroll his real estate ambitions. Early this week, he announced that Singapore-listed SingHaiyi Group of which he is a non-government chairman, had outbid contenders at a trustee's public sale to bag a San Jose industrial condominium challenge referred to as "Vietnam City" at what some would name a dust low-cost price of $33 million.<br><br>Nevertheless it should be said that I could always be unsuitable. Relating to investing, we can't afford to be overly assured. We have to be conscious of the potential of being mistaken. So even if property [http://www.opensource4arab.com/node/98312 condo prices in singapore] rise as a substitute of fall, I would solely make much less cash by not shopping for extra properties now, which is okay by me. I even have a friend who bought a business property near Tai Seng MRT station in 2010, the place its location is obviously relatively convenient. The property's momentary occupation permit (HIGH) was in May 2011, however even after a couple of months, he nonetheless has not discovered a tenant. This is in spite of the theoretical rental yield of about 5 per cent to 6 per cent primarily based on current property costs. Provide challenge co-ordination for property redevelopment Industrial<br><br>Actually, on 2 Might 2000, lawyers appearing for PPPL within the sale of Parkway Parade to Lend Lease took the place that there was no concluded contract between the parties and so they instructed the legal professionals of Lend Lease so. In February 2000, the primary defendants bought their interest in Parkway Parade. The plaintiffs have been informed of the sale in March 2000. With the sale, all negotiations between the plaintiffs and Parkway on the supposed lease effectively ceased. In July 2000, the plaintiffs indicated to the brand new homeowners their resolution to withdraw from the cineplex project. Li Zhiwei, a forestry worker in China, found this 19-inch earthworm in a gutter near his house. He plans to raise it as a pet. of internet admissions income versus base hire whichever is greater<br><br>As early as 1994, the parties to this action have been in discussions for the development of the cineplex. Initially, the proposal was for the plaintiffs to undertake the construction of the cineplex and thereafter lease the cineplex for 21 years from the second defendants. Some years later, owing to changes in circumstances, the plaintiffs decided not to undertake your complete challenge. Further negotiations for a revised development scheme, the place the plaintiffs would only lease the cineplex, started in March 1999. Negotiations between the events got here to an end with the sale of the primary defendants' interest in Parkway Parade in February 2000. In the long run, no lease was signed.<br><br>alternatives of property purchases, sales and leases. We warmly invite you to discover our virtual marketplace, specifically created to use the most recent advances in digital photographic technology to showcase properties in probably the most thrilling and visible manner. Our dedication to providing the perfect setting for buyers, sellers and real property professionals alike, will enable you to successfully negotiate by means of every facet of a property transaction. HubPages and Hubbers (authors) could earn revenue on this page based on affiliate relationships and advertisements with companions including Amazon, eBay, Google, and others. Serangoon / Thomson (D19-20) KSL D'Esplanade Condominium Johor Bahru, Malaysia for Gross sales! L3006936G RAY INTERNATIONAL PROPERTY CONSULTANTS PTE LTD<br><br>Singapore has additionally developed strong IP arbitration capabilities. WIPO has established an Arbitration and Mediation Centre (AMC) in Singapore, its solely centre outdoors Geneva, to support IP dispute resolution in Asia. The collaboration framework between the IPOS and the WIPO AMC permits events to settle IP disputes via different dispute resolution avenues at the WIPO AMC. The Institute of Estate Agents (IEA) will take away its pointers on property agents' commissions next month, to fall in keeping with the Competition Act. In line with newest official figures, there has additionally been little upward motion in the non-public property rental market. Balestier / Geylang (D12-14) MODEL NEW APT, RITZ @ FARRER (RACE COURSE STREET) FOR SALE. CALL 9 Frasers Business Trust (FCOT) |
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| If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. The simplest counterexample is the [[harmonic series (mathematics)|harmonic series]]
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| :<math>1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots =\sum_{n=1}^\infty\frac{1}{n}.</math>
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| The divergence of the harmonic series [[Harmonic series (mathematics)#Divergence|was proven]] by the medieval mathematician [[Nicole Oresme]].
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| In specialized mathematical contexts, values can be usefully assigned to certain series whose sequence of partial sums diverges. A '''summability method''' or '''summation method''' is a [[partial function]] from the set of sequences of partial sums of series to values. For example, [[Cesàro summation]] assigns [[Grandi's series|Grandi's divergent series]]
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| :<math>1 - 1 + 1 - 1 + \cdots</math> | |
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| the value <sup>1</sup>/<sub>2</sub>. Cesàro summation is an '''averaging''' method, in that it relies on the [[arithmetic mean]] of the sequence of partial sums. Other methods involve [[analytic continuation]]s of related series. In [[physics]], there are a wide variety of summability methods; these are discussed in greater detail in the article on [[regularization (physics)|regularization]].
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| ==Theorems on methods for summing divergent series==
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| A summability method ''M'' is '''[[#Properties of summation methods|regular]]''' if it agrees with the actual limit on all [[convergent series]]. Such a result is called an '''[[abelian theorem]]''' for ''M'', from the prototypical [[Abel's theorem]]. More interesting and in general more subtle are partial converse results, called '''[[tauberian theorems]]''', from a prototype proved by [[Alfred Tauber]]. Here ''partial converse'' means that if ''M'' sums the series Σ, and some side-condition holds, then Σ was convergent in the first place; without any side condition such a result would say that ''M'' only summed convergent series (making it useless as a summation method for divergent series).
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| The operator giving the sum of a convergent series is '''[[#Properties of summation methods|linear]]''', and it follows from the [[Hahn–Banach theorem]] that it may be extended to a summation method summing any series with bounded partial sums. This fact is not very useful in practice since there are many such extensions, [[inconsistent]] with each other, and also since proving such operators exist requires invoking the [[axiom of choice]] or its equivalents, such as [[Zorn's lemma]]. They are therefore nonconstructive.
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| The subject of divergent series, as a domain of [[mathematical analysis]], is primarily concerned with explicit and natural techniques such as [[Abel summation]], [[Cesàro summation]] and [[Borel summation]], and their relationships. The advent of [[Wiener's tauberian theorem]] marked an epoch in the subject, introducing unexpected connections to [[Banach algebra]] methods in [[Fourier analysis]].
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| Summation of divergent series is also related to [[extrapolation]] methods and [[sequence transformation]]s as numerical techniques. Examples for such techniques are [[Padé approximant]]s, [[Levin-type sequence transformation]]s, and order-dependent mappings related to [[renormalization]] techniques for large-order [[perturbation theory]] in [[quantum mechanics]].
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| ==Properties of summation methods==
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| Summation methods usually concentrate on the sequence of partial sums of the series. While this sequence does not converge, we may often find that when we take an [[average]] of larger and larger initial terms of the sequence, the average converges, and we can use this average instead of a limit to evaluate the sum of the series. So in evaluating {{nowrap|1=''a'' = ''a''<sub>0</sub> + ''a''<sub>1</sub> + ''a''<sub>2</sub> + ...,}} we work with the sequence ''s'', where {{nowrap|1=''s''<sub>0</sub> = ''a''<sub>0</sub>}} and {{nowrap|1=''s''<sub>''n''+1</sub> = ''s''<sub>''n''</sub> + ''a''<sub>''n+1''</sub>}}. In the convergent case, the sequence ''s'' approaches the limit ''a''. A '''summation method''' can be seen as a function from a set of sequences of partial sums to values. If '''A''' is any summation method assigning values to a set of sequences, we may mechanically translate this to a '''series-summation method''' '''A'''<sup>Σ</sup> that assigns the same values to the corresponding series. There are certain properties it is desirable for these methods to possess if they are to arrive at values corresponding to limits and sums, respectively.
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| # '''Regularity'''. A summation method is ''regular'' if, whenever the sequence ''s'' converges to ''x'', {{nowrap|1='''A'''(''s'') = ''x''.}} Equivalently, the corresponding series-summation method evaluates {{nowrap|1='''A'''<sup>Σ</sup>(''a'') = ''x''.}}
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| # '''Linearity'''. '''A''' is ''linear'' if it is a linear functional on the sequences where it is defined, so that {{nowrap|1 = '''A'''(''k'' ''r'' + ''s'') = ''k'' '''A'''(''r'') + '''A'''(''s'')}} for sequences ''r'', ''s'' and a real or complex scalar ''k''. Since the terms {{nowrap|1 = ''a''<sub>''n''</sub> = ''s''<sub>''n''+1</sub> − ''s''<sub>''n''</sub>}} of the series ''a'' are linear functionals on the sequence ''s'' and vice versa, this is equivalent to '''A'''<sup>Σ</sup> being a linear functional on the terms of the series.
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| # '''Stability'''. If ''s'' is a sequence starting from ''s''<sub>0</sub> and ''s''′ is the sequence obtained by omitting the first value and subtracting it from the rest, so that {{nowrap|1=''s''′<sub>''n''</sub> = ''s''<sub>''n''+1</sub> − ''s''<sub>0</sub>}}, then '''A'''(''s'') is defined if and only if '''A'''(''s''′) is defined, and {{nowrap|1='''A'''(''s'') = ''s''<sub>0</sub> + '''A'''(''s''′).}} Equivalently, whenever {{nowrap|1=''a''′<sub>''n''</sub> = ''a''<sub>''n''+1</sub>}} for all ''n'', then {{nowrap|1='''A'''<sup>Σ</sup>(''a'') = ''a''<sub>0</sub> + '''A'''<sup>Σ</sup>(''a''′).}}<ref>see Michon's Numericana http://www.numericana.com/answer/sums.htm</ref><ref>see also Translativity at The Encyclopedia of Mathematics wiki (Springer) [http://www.encyclopediaofmath.org/index.php/Translativity_of_a_summation_method]</ref>
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| The third condition is less important, and some significant methods, such as [[Borel summation]], do not possess it.{{Citation needed|reason=I have been unable to find further information about this|date=July 2013}}
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| One can also give a weaker alternative to the last condition.
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| # '''Finite Re-indexability'''. If ''s'' and ''s''′ are two sequences such that there exists a [[bijection]] <math> f: \mathbb{N} \rightarrow \mathbb{N} </math> such that {{nowrap|1=''s''<sub>''i''</sub> = ''s''′<sub>''f(i)''</sub>}} for all ''i'', and if there exists some <math> N \in \mathbb{N} </math> such that {{nowrap|1=''s''<sub>''i''</sub> = ''s''′<sub>''i''</sub>}} for all ''i'' > ''N'', then {{nowrap|1='''A'''(''s'') = '''A'''(''s''′).}} (In other words, ''s''′ is the same sequence as ''s'', with only finitely many terms re-indexed.) Note that this is a weaker condition than '''Stability''', because any summation method that exhibits '''Stability''' also exhibits '''Finite Re-indexability''', but the converse is not true.
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| A desirable property for two distinct summation methods '''A''' and '''B''' to share is ''consistency'': '''A''' and '''B''' are [[consistent]] if for every sequence ''s'' to which both assign a value, {{nowrap|1 = '''A'''(''s'') = '''B'''(''s'').}} If two methods are consistent, and one sums more series than the other, the one summing more series is ''stronger''.
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| There are powerful numerical summation methods that are neither regular nor linear, for instance nonlinear [[sequence transformation]]s like [[Levin-type sequence transformation]]s and [[Padé approximant]]s, as well as the order-dependent mappings of perturbative series based on [[renormalization]] techniques.
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| ==Axiomatic methods==
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| Taking regularity, linearity and stability as axioms, it is possible to sum many divergent series by elementary algebraic manipulations. For instance, whenever {{nowrap|1=''r'' ≠ 1,}} the [[Divergent geometric series|geometric series]]
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| :<math>\begin{align}
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| G(r,c) & = \sum_{k=0}^\infty cr^k & & \\
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| & = c + \sum_{k=0}^\infty cr^{k+1} & & \mbox{ (stability) } \\
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| & = c + r \sum_{k=0}^\infty cr^k & & \mbox{ (linearity) } \\
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| & = c + r \, G(r,c), & & \mbox{ whence } \\
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| G(r,c) & = \frac{c}{1-r} ,\mbox{unless it is infinite} & & \\
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| \end{align}</math>
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| can be evaluated regardless of convergence. More rigorously, any summation method that possesses these properties and which assigns a finite value to the geometric series must assign this value. However, when ''r'' is a real number larger than 1, the partial sums increase without bound, and averaging methods assign a limit of ∞.
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| ==Nørlund means==
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| Suppose ''p<sub>n</sub>'' is a sequence of positive terms, starting from ''p''<sub>0</sub>. Suppose also that
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| :<math>\frac{p_n}{p_0+p_1 + \cdots + p_n} \rightarrow 0.</math>
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| If now we transform a sequence s by using ''p'' to give weighted means, setting
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| :<math>t_m = \frac{p_m s_0 + p_{m-1}s_1 + \cdots + p_0 s_m}{p_0+p_1+\cdots+p_m}</math>
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| then the limit of ''t<sub>n</sub>'' as ''n'' goes to infinity is an average called the '''[[Niels Erik Nørlund|Nørlund]] mean''' '''N'''<sub>''p''</sub>(''s'').
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| The Nørlund mean is regular, linear, and stable. Moreover, any two Nørlund means are consistent. The most significant of the Nørlund means are the Cesàro sums. Here, if we define the sequence ''p<sup>k</sup>'' by
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| :<math>p_n^k = {n+k-1 \choose k-1}</math>
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| then the Cesàro sum C<sub>''k''</sub> is defined by {{nowrap|1=C<sub>''k''</sub>(''s'') = '''N'''<sub><big>(</big>''p<sup>k</sup>''<big>)</big></sub>(''s'').}} Cesàro sums are Nørlund means if {{nowrap|1=''k'' ≥ 0}}, and hence are regular, linear, stable, and consistent. C<sub>0</sub> is ordinary summation, and C<sub>1</sub> is ordinary [[Cesàro summation]]. Cesàro sums have the property that if {{nowrap|1=''h'' > ''k'',}} then C<sub>''h''</sub> is stronger than C<sub>''k''</sub>.
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| ==Abelian means==
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| Suppose λ = {λ<sub>''0''</sub>, λ<sub>''1''</sub>, λ<sub>''2''</sub>, ...} is a strictly increasing sequence tending towards infinity, and that {{nowrap|1=λ<sub>0</sub> ≥ 0}}. Suppose
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| :<math>f(x) = \sum_{n=0}^\infty a_n \exp(-\lambda_n x)</math>
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| converges for all real numbers ''x''>0. Then the '''Abelian mean''' ''A''<sub>λ</sub> is defined as
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| :<math>A_\lambda(s) = \lim_{x \rightarrow 0^{+}} f(x).</math>
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| More generally, if the series for ''f'' only converges for large ''x'' but can be analytically continued to all positive real ''x'', then one can still define the sum of the divergent series by the limit above.
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| A series of this type is known as a generalized [[Dirichlet series]]; in applications to physics, this is known as the method of [[heat-kernel regularization]].
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| Abelian means are regular and linear, but not stable and not always consistent between different choices of λ. However, some special cases are very important summation methods.
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| ===Abel summation===
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| {{see also|Abel's theorem}}
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| If {{nowrap|1 = λ<sub>''n''</sub> = ''n''}}, then we obtain the method of '''Abel summation'''. Here
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| :<math>f(x) = \sum_{n=0}^\infty a_n e^{-nx} = \sum_{n=0}^\infty a_n z^n,</math>
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| where ''z'' = exp(−''x''). Then the limit of ƒ(''x'') as ''x'' approaches 0 through positive reals is the limit of the power series for ƒ(''z'') as ''z'' approaches 1 from below through positive reals, and the Abel sum ''A''(''s'') is defined as
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| :<math>A(s) = \lim_{z \rightarrow 1^{-}} \sum_{n=0}^\infty a_n z^n.</math>
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| Abel summation is interesting in part because it is consistent with but more powerful than [[Cesàro summation]]: {{nowrap|1 = ''A''(''s'') = ''C''<sub>''k''</sub>(''s'')}} whenever the latter is defined. The Abel sum is therefore regular, linear, stable, and consistent with Cesàro summation.
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| ===Lindelöf summation===
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| If {{nowrap|1 = λ<sub>''n''</sub> = ''n'' log(''n'')}}, then (indexing from one) we have
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| :<math>f(x) = a_1 + a_2 2^{-2x} + a_3 3^{-3x} + \cdots .</math>
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| Then ''L''(''s''), the '''Lindelöf sum''' {{harv|Volkov|2001}}, is the limit of ƒ(''x'') as ''x'' goes to zero. The Lindelöf sum is a powerful method when applied to power series among other applications, summing power series in the [[Mittag-Leffler star]].
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| If ''g''(''z'') is analytic in a disk around zero, and hence has a [[Maclaurin series]] ''G''(''z'') with a positive radius of convergence, then {{nowrap|1 = ''L''(''G''(''z'')) = ''g''(''z'')}} in the Mittag-Leffler star. Moreover, convergence to ''g''(''z'') is uniform on compact subsets of the star.
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| ==Moment methods==
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| Suppose that dμ is a measure on the non-negative real line such that all the moments
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| :<math>\mu_n=\int_0^\infty x^n d\mu</math>
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| are finite.
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| If ''a''<sub>0</sub>+''a''<sub>1</sub>+... is a series such that
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| :<math>a(x)=\frac{a_0x^0}{\mu_0}+\frac{a_1x^1}{\mu_1}+\cdots</math>
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| converges for all non-negative ''x'', then the (dμ) sum of the series is defined to be the value of the integral
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| :<math>\int_0^\infty a(x)d\mu</math>
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| if it is defined. (Note that if the numbers μ<sub>''n''</sub> increase too rapidly then they do not uniquely determine the measure μ.)
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| For example, if dμ = ''e''<sup>–''x''</sup>''dx'' then μ<sub>''n''</sub> = ''n''!, and this gives one version of [[Borel summation]].
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| ==Nonlinear methods==
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| ===Zeta function regularization===
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| If the series
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| :<math>f(s) = \frac{1}{a_1^s} + \frac{1}{a_2^s} + \frac{1}{a_3^s}+ \cdots </math>
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| converges for large real ''s'' and can be analytically continued along the real line to ''s''=–1, then its value at ''s''=–1 is called the zeta regularized sum of the series ''a''<sub>1</sub>+''a''<sub>2</sub>+... In applications, the numbers ''a''<sub>''i''</sub> are sometimes the eigenvalues of a self-adjoint operator ''A'' with compact resolvant, and ''f''(''s'') is then the trace of ''A''<sup>–''s''</sub>. For example, if ''A'' has eigenvalues 1, 2, 3, ... then ''f''(''s'') is the Riemann zeta function, whose value at ''s''=–1 is –1/12.
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| ==See also==
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| * [[1 − 1 + 2 − 6 + 24 − 120 + · · ·]]
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| * [[Borel summation]]
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| * [[Zeta function regularization]]
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| * [[Euler summation]]
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| * [[Lambert summation]]
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| * [[Silverman–Toeplitz theorem]]
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| * [[Hölder summation]]
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| ==References==
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| {{reflist}}
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| * {{citation|title=Large-Order Perturbation Theory and Summation Methods in Quantum Mechanics|first1=G.A.|last1=Arteca|first2=F.M.|last2=Fernández|first3=E.A.|last3=Castro|publisher=Springer-Verlag|publication-place=Berlin|year=1990}}.
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| * {{citation|title=Padé Approximants|first1=G. A.|last1=Baker, Jr.|first2=P. |last2=Graves-Morris|publisher=Cambridge University Press|year=1996}}.
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| * {{citation|title=Extrapolation Methods. Theory and Practice|first1=C.|last1=Brezinski|first2=M. Redivo|last2=Zaglia|publisher=North-Holland|year=1991}}.
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| * {{citation|title=Divergent Series|first=G. H.|last=Hardy|authorlink=G. H. Hardy|publication-place=Oxford|publisher=Clarendon Press|year=1949|url=http://www.archive.org/details/divergentseries033523mbp}}.
| |
| * {{citation|title=Large-Order Behaviour of Perturbation Theory|first1=J.-C.|last1=LeGuillou|first2=J.|last2=Zinn-Justin|publisher=North-Holland|publication-place=Amsterdam|year=1990}}.
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| * {{springer|id=l/l058990|title=Lindelöf summation method|first=I.I.|last=Volkov|year=2001}}.
| |
| * {{springer|id=a/a010170|title=Abel summation method|first=A.A.|last=Zakharov|year=2001}}.
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| {{Series (mathematics)}}
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| [[Category:Divergent series]]
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| [[Category:Mathematical series]]
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| [[Category:Summability methods]]
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| [[Category:Asymptotic analysis]]
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| [[Category:Summability theory]]
| |
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Nevertheless it should be said that I could always be unsuitable. Relating to investing, we can't afford to be overly assured. We have to be conscious of the potential of being mistaken. So even if property condo prices in singapore rise as a substitute of fall, I would solely make much less cash by not shopping for extra properties now, which is okay by me. I even have a friend who bought a business property near Tai Seng MRT station in 2010, the place its location is obviously relatively convenient. The property's momentary occupation permit (HIGH) was in May 2011, however even after a couple of months, he nonetheless has not discovered a tenant. This is in spite of the theoretical rental yield of about 5 per cent to 6 per cent primarily based on current property costs. Provide challenge co-ordination for property redevelopment Industrial
Actually, on 2 Might 2000, lawyers appearing for PPPL within the sale of Parkway Parade to Lend Lease took the place that there was no concluded contract between the parties and so they instructed the legal professionals of Lend Lease so. In February 2000, the primary defendants bought their interest in Parkway Parade. The plaintiffs have been informed of the sale in March 2000. With the sale, all negotiations between the plaintiffs and Parkway on the supposed lease effectively ceased. In July 2000, the plaintiffs indicated to the brand new homeowners their resolution to withdraw from the cineplex project. Li Zhiwei, a forestry worker in China, found this 19-inch earthworm in a gutter near his house. He plans to raise it as a pet. of internet admissions income versus base hire whichever is greater
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Singapore has additionally developed strong IP arbitration capabilities. WIPO has established an Arbitration and Mediation Centre (AMC) in Singapore, its solely centre outdoors Geneva, to support IP dispute resolution in Asia. The collaboration framework between the IPOS and the WIPO AMC permits events to settle IP disputes via different dispute resolution avenues at the WIPO AMC. The Institute of Estate Agents (IEA) will take away its pointers on property agents' commissions next month, to fall in keeping with the Competition Act. In line with newest official figures, there has additionally been little upward motion in the non-public property rental market. Balestier / Geylang (D12-14) MODEL NEW APT, RITZ @ FARRER (RACE COURSE STREET) FOR SALE. CALL 9 Frasers Business Trust (FCOT)