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| {| border="1" style="float: right; border-collapse: collapse; margin: 0 0 0 1em;"
| | Various modes of transport can be found close to Kallang Riverside condominium together with buying centers and restaurants. Kallang Riverside apartment is also just mins away from the Lavendar Mrt Station and within the neighborhood of Stadium MRT, Kallang MRT. Commuting to Orchard in addition to the town/Marina Bay space is subsequently very handy.<br><br>No danger of construction delays. Completed properties are simply that. They're completed and prepared so that you can play the position of landlord. Whereas there at all times a [http://econo.urin79.com/?document_srl=1910146 Econo.Urin79.Com] risk of new launches not assembly their deadlines irrespective of how small the chance is. In bad situations, development could also be delayed by 6 months. In worst scenarios, builders can go bust. Don't assume that can occur in Singapore? Read up about the national stadium , motorsports hub , and oil rig accident among others.<br><br>An opulent and fashionable lifestyle awaits you at Inflora condo which is a new launch Pasir Ris that encompasses amazing amenities / amenities and wonderful scorching spots of Singapore. Inflora Condo is constructed near to the upcoming Tampines East MRT station which is simply strolling distance away, making this flora highway condo much more worthwhile. This Pasir Ris rental consists of services resembling, waterjet pool, swimming pool, club house, dining pavilion and plenty of more. Simply at Inflora rental alone, you and your family members can enjoy enjoyable-crammed day all the best way to the night.<br><br>Many buyers feel that the question now is just not whether or not you ought to be investing in properties. However whether or not to buy an underneath building condominium or purchase a completed resale property for investment. This superb Pasir Ris Rental have all the facilities and amenities which you'll ever want with so many exciting actions you can get pleasure from. Inflora apartment the brand new launch at Pasir Ris, is the perfect dwelling for anyone who is on the lookout for a contemporary and classy lifestyle. Register your interest with us, to obtain prime priority in receiving the Inflora condominium latest updates and to visit our showflat first-hand when is open to public. The query then is, is that this the appropriate time to purchase now? Will costs come down?<br><br>It's theoretically possible to get a distinct loan from a special lender for each fee. However doing that is illogical and possibly quite a hassle. The vital point to notice is that since your payments to the developer is progressive, the mortgage housing loan that you take up can also be disbursed portion by portion. This means that the money is release slowly to match the agreed cost schedule. When this is taken into account, you might only be beginning to service your loan 1 yr after your purchase. The installments that you'll repay may even be based mostly on the quantity which have been released.<br><br>Firstly, it is important to understand that investors view a completed resale property is of a lesser risk than one that's underneath construction. This view is also shared by the banks as you will usually find that the very best mortgage charges obtainable are for accomplished properties. Decrease threat means lower rates of interest. So unless you have a very particular relationship with a financial institution of monetary establishment, the most effective housing loans that you can get will almost definitely be the best loan bundle for underneath development property, not the perfect deal out there.<br><br>Several buses can be found on the bus interchange near Skyvue condo along with shopping facilities and restaurants. Skyvue condo can be simply 3 mins away from the Bishan MRT interchange Station (CC15,NS17) for both the circle and the north-south strains. Commuting to Orchard as well as the town area is due to this fact very handy. Additionally it is near to good malls like Junction eight, Nex megamall, Ang mo kio hub, Bishan Park and Toa Payoh Central Leisure in your loved ones and mates are due to this fact at your fingertips with the complete condo amenities as well as the amenities close by. Residents get to get pleasure from numerous enjoyable-filling nearby recreation actions. |
| | colspan="2" align="center" | {{Irrational numbers}}
| |
| |-
| |
| |[[Binary numeral system|Binary]]
| |
| | {{gaps|1.00110|01110|11101...}}
| |
| |-
| |
| | [[Decimal]]
| |
| | {{gaps|1.20205|69031|59594|2854...}}
| |
| |-
| |
| | [[Hexadecimal]]
| |
| | {{gaps|1.33BA0|04F00|621383...}}
| |
| |-
| |
| | [[Continued fraction]]
| |
| | <math>1 + \frac{1}{4 + \cfrac{1}{1 + \cfrac{1}{18 + \cfrac{1}{\ddots\qquad{}}}}}</math><br><small>Note that this continuing fraction is not [[Periodic continued fraction|periodic]].</small>
| |
| |}
| |
| | |
| In [[mathematics]], '''Apéry's constant''' is a number that occurs in a variety of situations. It arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's [[gyromagnetic ratio]] using quantum electrodynamics. It also arises in conjunction with the [[gamma function]] when solving certain integrals involving exponential functions in a quotient which appear occasionally in physics, for instance when evaluating the two-dimensional case of the [[Debye model]] and the [[Stefan–Boltzmann law]].
| |
| It is defined as the number ζ(3),
| |
| | |
| :<math>\zeta(3)=\sum_{k=1}^\infty\frac{1}{k^3}=1+\frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + \frac{1}{6^3} + \frac{1}{7^3} + \frac{1}{8^3} + \frac{1}{9^3} + \cdots\,\!</math>
| |
| | |
| where ζ is the [[Riemann zeta function]]. It has an approximate value of {{harv|Wedeniwski|2001}}
| |
| | |
| :ζ(3) = {{gaps|1.20205|69031|59594|28539|97381|61511|44999|07649|86292...}} {{OEIS|id=A002117}}.
| |
| | |
| The [[reciprocal (mathematics)|reciprocal]] of this [[constant (mathematics)|constant]] is the [[probability]] that any three [[positive integer]]s, chosen at random, will be [[relatively prime]] (in the sense that as ''N ''goes to infinity, the probability that three positive integers less than ''N'' chosen uniformly at random will be relatively prime approaches this value).
| |
| | |
| ==Apéry's theorem==
| |
| {{main|Apéry's theorem}}
| |
| | |
| This value was named for [[Roger Apéry]] (1916–1994), who in 1978 proved it to be [[irrational number|irrational]]. This result is known as ''[[Apéry's theorem]]''. The original proof is complex and hard to grasp, and shorter proofs have been found later, using [[Legendre polynomials]]. It is not known whether Apéry's constant is [[transcendental number|transcendental]].
| |
| | |
| Work by [[Wadim Zudilin]] and Tanguy Rivoal has shown that infinitely many of the numbers ζ(2''n''+1) must be irrational,<ref>{{Citation |author=T. Rivoal |title=La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs |journal=Comptes Rendus de l'Académie des Sciences. Série I. Mathématique |volume=331 |year=2000 |pages=267–270 |postscript=.|doi=10.1016/S0764-4442(00)01624-4}}</ref> and even that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.<ref>{{Citation |author=W. Zudilin |title=One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational |journal=Russ. Math. Surv. |year=2001 |volume=56 |pages=774–776 |doi=10.1070/RM2001v056n04ABEH000427 |postscript=. |issue=4}}</ref>
| |
| | |
| ==Series representation==
| |
| In 1772, [[Leonhard Euler]] {{harv|Euler|1773}} gave the series representation {{harv|Srivastava|2000|loc=p. 571 (1.11)}}:
| |
| | |
| :<math>\zeta(3)=\frac{\pi^2}{7}
| |
| \left[ 1-4\sum_{k=1}^\infty \frac {\zeta (2k)} {(2k+1)(2k+2) 2^{2k}} \right]</math>
| |
| | |
| which was subsequently rediscovered several times.
| |
| | |
| [[Ramanujan]] gives several series, which are notable in that they can provide several digits of accuracy per iteration. These include:<ref>Bruce C. Berndt, ''Ramanujan's notebooks, Part II'' (1989), Springer-Verlag. ''See chapter 14, formulas 25.1 and 25.3''</ref>
| |
| | |
| :<math>\zeta(3)=\frac{7}{180}\pi^3 -2
| |
| \sum_{k=1}^\infty \frac{1}{k^3 (e^{2\pi k} -1)}</math>
| |
| | |
| [[Simon Plouffe]] has developed other series {{harv|Plouffe|1998}}:
| |
| | |
| :<math>\zeta(3)= 14
| |
| \sum_{k=1}^\infty \frac{1}{k^3 \sinh(\pi k)}
| |
| -\frac{11}{2}
| |
| \sum_{k=1}^\infty \frac{1}{k^3 (e^{2\pi k} -1)}
| |
| -\frac{7}{2}
| |
| \sum_{k=1}^\infty \frac{1}{k^3 (e^{2\pi k} +1)}.
| |
| </math>
| |
| | |
| Similar relations for the values of <math>\zeta(2n+1)</math> are given in the article [[zeta constants]].
| |
| | |
| Many additional series representations have been found, including:
| |
| | |
| :<math>\zeta(3) = \frac{8}{7} \sum_{k=0}^\infty \frac{1}{(2k+1)^3}</math>
| |
| | |
| :<math>\zeta(3) = \frac{4}{3} \sum_{k=0}^\infty \frac{(-1)^k}{(k+1)^3}</math>
| |
| | |
| :<math>\zeta(3) = \frac{5}{2} \sum_{k=1}^\infty (-1)^{k-1} \frac{k!^2}{k^3 (2k)!}</math>
| |
| | |
| :<math>\zeta(3) = \frac{1}{4} \sum_{k=1}^\infty (-1)^{k-1}
| |
| \frac{56k^2-32k+5}{(2k-1)^2} \frac{(k-1)!^3}{(3k)!}</math>
| |
| | |
| :<math>\zeta(3)=\frac{8}{7}-\frac{8}{7}\sum_{k=1}^\infty \frac{{\left( -1 \right) }^k\,2^{-5 + 12\,k}\,k\,
| |
| \left( -3 + 9\,k + 148\,k^2 - 432\,k^3 - 2688\,k^4 + 7168\,k^5 \right) \,
| |
| {k!}^3\,{\left( -1 + 2\,k \right) !}^6}{{\left( -1 + 2\,k \right) }^3\,
| |
| \left( 3\,k \right) !\,{\left( 1 + 4\,k \right) !}^3}</math>
| |
| | |
| :<math>\zeta(3) = \sum_{k=0}^\infty (-1)^k \frac{205k^2 + 250k + 77}{64} \frac{k!^{10}}{(2k+1)!^5}</math>
| |
| | |
| and
| |
| | |
| :<math>\zeta(3) = \sum_{k=0}^\infty (-1)^k \frac{P(k)}{24}
| |
| \frac{((2k+1)!(2k)!k!)^3}{(3k+2)!(4k+3)!^3}</math>
| |
| | |
| where
| |
| | |
| :<math>P(k) = 126392k^5 + 412708k^4 + 531578k^3 + 336367k^2 + 104000k + 12463.\,</math>
| |
| | |
| Some of these have been used to calculate Apéry's constant with several million digits.
| |
| | |
| {{harv|Broadhurst|1998}} gives a series representation that allows arbitrary [[binary digit]]s to be computed, and thus, for the constant to be obtained in nearly [[linear time]], and [[logarithmic space]].
| |
| | |
| ==Other formulas==
| |
| Apéry's constant can be expressed in terms of the second-order [[polygamma function]] as
| |
| | |
| :<math>\zeta(3) = -\frac{1}{2} \, \psi^{(2)}(1).</math>
| |
| | |
| It can be expressed as the infinite [[continued fraction]] [1; 4, 1, 18, 1, 1, 1, 4, 1, ...] {{OEIS|id=A013631}}.
| |
| | |
| ==Known digits==
| |
| | |
| The number of known digits of Apéry's constant ζ(3) has increased dramatically during the last decades. This is due both to the increase of performance of computers and to algorithmic improvements.
| |
| | |
| {| class="wikitable" style="margin: 1em auto 1em auto"
| |
| |+ '''Number of known decimal digits of Apéry's constant ζ(3)'''
| |
| ! Date || Decimal digits || Computation performed by
| |
| |-
| |
| | 1735 || 16 || [[Leonhard Euler]]
| |
| |-
| |
| | unknown || 16 || [[Adrien-Marie Legendre]]
| |
| |-
| |
| | 1887 || 32 || [[Thomas Joannes Stieltjes]]
| |
| |-
| |
| | 1996 || 520,000 || Greg J. Fee & [[Simon Plouffe]]
| |
| |-
| |
| | 1997 || 1,000,000 || Bruno Haible & Thomas Papanikolaou
| |
| |-
| |
| | May 1997 || 10,536,006 || Patrick Demichel
| |
| |-
| |
| | February 1998 || 14,000,074 || Sebastian Wedeniwski
| |
| |-
| |
| | March 1998 || 32,000,213 || Sebastian Wedeniwski
| |
| |-
| |
| | July 1998 || 64,000,091 || Sebastian Wedeniwski
| |
| |-
| |
| | December 1998 || 128,000,026 || Sebastian Wedeniwski {{harv|Wedeniwski|2001}}
| |
| |-
| |
| | September 2001 || 200,001,000 || Shigeru Kondo & Xavier Gourdon
| |
| |-
| |
| | February 2002 || 600,001,000 || Shigeru Kondo & Xavier Gourdon
| |
| |-
| |
| | February 2003 || 1,000,000,000 || Patrick Demichel & Xavier Gourdon
| |
| |-
| |
| | April 2006 || 10,000,000,000 || Shigeru Kondo & Steve Pagliarulo (see {{harvtxt|Gourdon|Sebah|2003}})
| |
| |-
| |
| | January 2009 || 15,510,000,000 || Alexander J. Yee & Raymond Chan (see {{harvtxt|Yee|Chan|2009}})
| |
| |-
| |
| | March 2009 || 31,026,000,000 || Alexander J. Yee & Raymond Chan (see {{harvtxt|Yee|Chan|2009}})
| |
| |-
| |
| | September 2010 || 100,000,001,000 || Alexander J. Yee (see [http://www.numberworld.org/digits/Zeta%283%29/ Yee])
| |
| |-
| |
| | September 2013 || 200,000,001,000 || Robert J. Setti ([http://settifinancial.com/01042-aperys-constant-zeta3-world-record-computation/ Aprey's Constant - Zeta(3) - 200 Billion Digits])
| |
| |}
| |
| | |
| ==Notes==
| |
| {{Reflist}}
| |
| | |
| ==References==
| |
| *{{citation
| |
| | last = Euler
| |
| | first = Leonhard
| |
| | authorlink = Leonhard Euler
| |
| | year = 1773
| |
| | title = Exercitationes analyticae
| |
| | journal = Novi Commentarii academiae scientiarum Petropolitanae
| |
| | volume = 17
| |
| | pages = 173–204
| |
| | url = http://math.dartmouth.edu/~euler/docs/originals/E432.pdf
| |
| | language = Latin
| |
| | format = PDF
| |
| | accessdate = 2008-05-18
| |
| }}
| |
| * {{cite news
| |
| |first=V.
| |
| |last=Ramaswami
| |
| |title=Notes on Riemann's ζ-function
| |
| |year=1934
| |
| |journal=J. London Math. Soc.
| |
| |volume=9
| |
| |pages=165–169
| |
| |doi=10.1112/jlms/s1-9.3.165|issue=3}}
| |
| * {{citation
| |
| |first=Roger
| |
| |last=Apéry
| |
| |title=Irrationalité de ζ(2) et ζ(3)
| |
| |year=1979
| |
| |journal=Astérisque|volume=61|pages=11–13}}
| |
| * {{Citation
| |
| |author=A. van der Poorten
| |
| |title=A proof that Euler missed..
| |
| |journal=[[The Mathematical Intelligencer]]
| |
| |volume=1
| |
| |year=1979
| |
| |pages=195–203
| |
| |doi=10.1007/BF03028234
| |
| |url=http://www.maths.mq.edu.au/~alf/45.pdf
| |
| |issue=4}}
| |
| * {{cite journal
| |
| |journal=El. J. Combinat
| |
| |year=1996
| |
| |volume=3
| |
| |first1=Tewodoros
| |
| |last1=Amdeberhan
| |
| |url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i1r13
| |
| |pages=#R13
| |
| |title=Faster and faster convergent series for ζ(3)
| |
| }}
| |
| *{{cite arXiv| first=D.J.| last=Broadhurst| eprint=math.CA/9803067| title=Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)| year=1998}}
| |
| * {{citation|first=Simon|last=Plouffe|url=http://www.lacim.uqam.ca/~plouffe/identities.html|title=Identities inspired from Ramanujan Notebooks II|year=1998}}
| |
| * {{citation|first=Simon|last=Plouffe|url=http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap97.html|title=Zeta(3) or Apery constant to 2000 places|year=undated}}
| |
| * {{citation|first=S.|last=Wedeniwski|title=The Value of Zeta(3) to 1,000,000 places|editor=Simon Plouffe|year=2001|publisher=Project Gutenberg}}
| |
| * {{citation
| |
| | last = Srivastava | first = H. M.
| |
| |date=December 2000
| |
| | title = Some Families of Rapidly Convergent Series Representations for the Zeta Functions
| |
| | url = http://www.math.nthu.edu.tw/~tjm/abstract/0012/tjm0012_3.pdf
| |
| | format = PDF
| |
| | journal = Taiwanese Journal of Mathematics
| |
| | volume = 4 | issue = 4 | pages = 569–598
| |
| | oclc =36978119
| |
| | accessdate = 2008-05-18
| |
| }}
| |
| * {{cite web
| |
| |first1=Xavier
| |
| |last1=Gourdon
| |
| |first2=Pascal
| |
| |last2=Sebah
| |
| |url=http://numbers.computation.free.fr/Constants/Zeta3/zeta3.html
| |
| |title=The Apéry's constant: z(3)
| |
| |year=2003}}
| |
| * {{mathworld|title=Apéry's constant|urlname=AperysConstant}}
| |
| * {{citation|first1=Alexander J.|last1=Yee|first2=Raymond|last2=Chan|url=http://www.numberworld.org/nagisa_runs/computations.html|title=Large Computations|year=2009}}
| |
| | |
| {{PlanetMath attribution|id=4021|title=Apéry's constant}}
| |
| | |
| {{DEFAULTSORT:Aperys constant}}
| |
| [[Category:Mathematical constants]]
| |
| [[Category:Analytic number theory]]
| |
| [[Category:Irrational numbers]]
| |
| [[Category:Zeta and L-functions]]
| |
Various modes of transport can be found close to Kallang Riverside condominium together with buying centers and restaurants. Kallang Riverside apartment is also just mins away from the Lavendar Mrt Station and within the neighborhood of Stadium MRT, Kallang MRT. Commuting to Orchard in addition to the town/Marina Bay space is subsequently very handy.
No danger of construction delays. Completed properties are simply that. They're completed and prepared so that you can play the position of landlord. Whereas there at all times a Econo.Urin79.Com risk of new launches not assembly their deadlines irrespective of how small the chance is. In bad situations, development could also be delayed by 6 months. In worst scenarios, builders can go bust. Don't assume that can occur in Singapore? Read up about the national stadium , motorsports hub , and oil rig accident among others.
An opulent and fashionable lifestyle awaits you at Inflora condo which is a new launch Pasir Ris that encompasses amazing amenities / amenities and wonderful scorching spots of Singapore. Inflora Condo is constructed near to the upcoming Tampines East MRT station which is simply strolling distance away, making this flora highway condo much more worthwhile. This Pasir Ris rental consists of services resembling, waterjet pool, swimming pool, club house, dining pavilion and plenty of more. Simply at Inflora rental alone, you and your family members can enjoy enjoyable-crammed day all the best way to the night.
Many buyers feel that the question now is just not whether or not you ought to be investing in properties. However whether or not to buy an underneath building condominium or purchase a completed resale property for investment. This superb Pasir Ris Rental have all the facilities and amenities which you'll ever want with so many exciting actions you can get pleasure from. Inflora apartment the brand new launch at Pasir Ris, is the perfect dwelling for anyone who is on the lookout for a contemporary and classy lifestyle. Register your interest with us, to obtain prime priority in receiving the Inflora condominium latest updates and to visit our showflat first-hand when is open to public. The query then is, is that this the appropriate time to purchase now? Will costs come down?
It's theoretically possible to get a distinct loan from a special lender for each fee. However doing that is illogical and possibly quite a hassle. The vital point to notice is that since your payments to the developer is progressive, the mortgage housing loan that you take up can also be disbursed portion by portion. This means that the money is release slowly to match the agreed cost schedule. When this is taken into account, you might only be beginning to service your loan 1 yr after your purchase. The installments that you'll repay may even be based mostly on the quantity which have been released.
Firstly, it is important to understand that investors view a completed resale property is of a lesser risk than one that's underneath construction. This view is also shared by the banks as you will usually find that the very best mortgage charges obtainable are for accomplished properties. Decrease threat means lower rates of interest. So unless you have a very particular relationship with a financial institution of monetary establishment, the most effective housing loans that you can get will almost definitely be the best loan bundle for underneath development property, not the perfect deal out there.
Several buses can be found on the bus interchange near Skyvue condo along with shopping facilities and restaurants. Skyvue condo can be simply 3 mins away from the Bishan MRT interchange Station (CC15,NS17) for both the circle and the north-south strains. Commuting to Orchard as well as the town area is due to this fact very handy. Additionally it is near to good malls like Junction eight, Nex megamall, Ang mo kio hub, Bishan Park and Toa Payoh Central Leisure in your loved ones and mates are due to this fact at your fingertips with the complete condo amenities as well as the amenities close by. Residents get to get pleasure from numerous enjoyable-filling nearby recreation actions.