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| In [[mathematics]], for a [[sequence]] of complex numbers ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... the '''infinite product'''
| | They have powerful and durable mountain bikes with a high-tech suspension system. Macaskill is tackling the Cuillin Ridge in Scotland, not far from where this two-wheeling genius hails from on the Isle of Skye. They both have some parts that are worse than another's, but it doesn't matter in general. its also important to look at the design of the bike some bikes are designed for specific porpuses that is to say , there bike s designed for sports , transport porpuses while other motobike are designed for prestige , for leisure porpuses before you go the market ldentify your porpuses why you want tobuy the mountai bike or any bike. Certain Haro mountain bike models worth mentioning right here consist of BMX Black trail X1, BMX Haro Partial 16, BMX Haro FIC and so forth. <br><br>If you want to get information about some really special and unique forks visit my site. When you have any issues about wherever and tips on how to employ [http://originsofttech.net/AgaArcade/profile/librain Choosing the right ride for you mountain bike sizing.], you can email us with the internet site. "Before breathing exercises it took me 45 minutes to really get into my stride when off road mountain biking. If you don't do a lot of racing on mountain trails then you need a road bike. - Get a bike that is lightweight and made out of carbon fiber materials. UST kits can also improve sturdiness of UST wheels with added sealing capacity in case of any punctures. <br><br>The steeper the angles, the more beneficial it would be for stability and high speed pedaling. If you do it as soon as you get home, then its done and you can concentrate on eating and relaxing. Guided by the dolphin trainer, they will perform a series of tricks such as clapping, synchronized jumping, etc. Being humiliated by my buddies was not fun, but there were enough good things about skiing that I wanted to come back for more, just not with my buddies. -Del Plomo Hot Springs: Ideal for bikes, low traffic. <br><br>Read product reviews and cycling magazines, research online, and ask for advice at your local bike shop. Cold and long winters at high altitude are perfect conditions for winter recreation. Article Source: mini bikes, minibikes and pocket rockets; At About-minibikes. Next, there is also a seat in the bike for the riders to sit during the riding. Cross country Nearly all of the mountain bikes available could be classed as cross country. <br><br>I ran with Hale and we also tried this method, it was effective. We have the best selection of bicycle parts( like wheels, bike handlebars, cycling shoes, bicycle pedals, bike cranks, shifters, bicycle brakes, bicycle tires and much more. Mountain biking amongst the majestic backdrop of the great mountains of Kerala is an unforgettable and thrilling experience. Suspension: Road bikes are built with a sole purpose of providing greater speed; they do not possess this feature, although they have certain materials which absorb the shocks of the uneven roads. They will provide you with a honest price, yet be ready to pay in between $650 and $4,000 on dual suspensions and in between $470 and $670 for hardtails. |
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| :<math>
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| \prod_{n=1}^{\infty} a_n = a_1 \; a_2 \; a_3 \cdots
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| </math>
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| is defined to be the [[limit of a sequence|limit]] of the partial products ''a''<sub>1</sub>''a''<sub>2</sub>...''a''<sub>''n''</sub> as ''n'' increases without bound. The product is said to ''[[Limit of a sequence|converge]]'' when the limit exists and is not zero. Otherwise the product is said to ''diverge''. A limit of zero is treated specially in order to obtain results analogous to those for [[Infinite series|infinite sums]]. Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. If the product converges, then the limit of the sequence ''a''<sub>''n''</sub> as ''n'' increases without bound must be 1, while the converse is in general not true.
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| The best known examples of infinite products are probably some of the formulae for [[pi|π]], such as the following two products, respectively by [[Viète]] ([[Viète's formula]], the first published infinite product in mathematics) and [[John Wallis]] ([[Wallis product]]):
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| :<math>\frac{2}{\pi} = \frac{ \sqrt{2} }{ 2 } \cdot \frac{ \sqrt{2 + \sqrt{2}} }{ 2 } \cdot \frac{ \sqrt{2 + \sqrt{2 + \sqrt{2}}} }{ 2 } \cdots</math>
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| :<math>\frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots = \prod_{n=1}^{\infty} \left( \frac{ 4 \cdot n^2 }{ 4 \cdot n^2 - 1 } \right). </math>
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| == Convergence criteria ==
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| The product of positive real numbers
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| :<math>\prod_{n=1}^{\infty} a_n</math>
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| converges if and only if the sum
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| :<math>\sum_{n=1}^{\infty} \log a_n</math>
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| converges. This allows the translation of convergence criteria for infinite sums into convergence criteria for infinite products. The same criterion applies to products of arbitrary complex numbers (including negative reals) if log is understood as a fixed [[Complex logarithm#Branches of the complex logarithm|branch of logarithm]] which satisfies log(1) = 0, with the proviso that the infinite product diverges when infinitely many ''a<sub>n</sub>'' fall outside the domain of log, whereas finitely many such ''a<sub>n</sub>'' can be ignored in the sum.
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| For products of reals in which each <math>a_n\ge1</math>, written as, for instance, <math>a_n=1+p_n</math>,
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| where <math>p_n\ge 0</math>, the bounds
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| :<math>1+\sum_{n=1}^{N} p_n \le \prod_{n=1}^{N} \left( 1 + p_n \right) \le \exp \left( \sum_{n=1}^{N}p_n \right)</math>
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| show that the infinite product converges precisely if the infinite sum of the ''p''<sub>''n''</sub> converges. This relies on the [[Monotone convergence theorem]]. More generally, the convergence of <math>\prod_{n=1}^\infty(1+p_n)</math> is equivalent to the convergence of <math>\sum_{n=1}^\infty p_n</math> if ''p<sub>n</sub>'' are real or complex numbers such that <math>\sum_{n=1}^\infty|p_n|^2<+\infty</math>, since <math>\log(1+x)=x+O(x^2)</math> in a neighbourhood of 0.
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| ==Product representations of functions==
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| {{main|Weierstrass factorization theorem}}
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| >One important result concerning infinite products is that every [[entire function]] ''f''(''z'') (that is, every function that is [[holomorphic function|holomorphic]] over the entire [[complex number|complex plane]]) can be factored into an infinite product of entire functions, each with at most a single root. In general, if ''f'' has a root of order ''m'' at the origin and has other complex roots at ''u''<sub>1</sub>, ''u''<sub>2</sub>, ''u''<sub>3</sub>, ... (listed with multiplicities equal to their orders), then
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| :<math>f(z) = z^m e^{\phi(z)} \prod_{n=1}^{\infty} \left(1 - \frac{z}{u_n} \right) \exp \left\lbrace \frac{z}{u_n} + \frac{1}{2}\left(\frac{z}{u_n}\right)^2 + \cdots + \frac{1}{\lambda_n} \left(\frac{z}{u_n}\right)^{\lambda_n} \right\rbrace </math>
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| where λ<sub>''n''</sub> are non-negative integers that can be chosen to make the product converge, and φ(''z'') is some uniquely determined analytic function (which means the term before the product will have no roots in the complex plane). The above factorization is not unique, since it depends on the choice of values for λ<sub>''n''</sub>, and is not especially elegant. However, for most functions, there will be some minimum non-negative integer ''p'' such that λ<sub>''n''</sub> = ''p'' gives a convergent product, called the [[Weierstrass_factorization_theorem|canonical product representation]]. This ''p'' is called the ''rank'' of the canonical product. <!-- The ''genus'' is the (is it max? or min?) of the degree of φ and ''p''. --> In the event that ''p'' = 0, this takes the form
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| :<math>f(z) = z^m e^{\phi(z)} \prod_{n=1}^{\infty} \left(1 - \frac{z}{u_n}\right).</math>
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| This can be regarded as a generalization of the [[Fundamental Theorem of Algebra]], since the product becomes finite and φ(''z'') is constant for polynomials.
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| In addition to these examples, the following representations are of special note:
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| {| cellspacing=15
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| |- valign=top
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| | [[Sine]] function
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| | <math>\sin(\pi z) = \pi z \prod_{n=1}^{\infty} \left(1 - \frac{z^2}{n^2}\right)</math>
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| | This is due to [[Euler]]. [[Wallis product|Wallis' formula for π]] is a special case of this.
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| |- valign=top
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| | [[Gamma function]]
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| | <math>\frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^{\infty} \left(1 + \frac{z}{n}\right) e^{-\frac{z}{n}}</math>
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| | [[Schlömilch]]
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| |- valign=top
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| | [[Weierstrass sigma function]]
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| | <math>\sigma(z) = z\prod_{\omega \in \Lambda_{*}} \left(1-\frac{z}{\omega}\right)e^{\frac{z^2}{2\omega^2}+\frac{z}{\omega}}</math>
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| | Here <math>\Lambda_{*}</math> is the lattice without the origin.
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| |- valign=top
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| | [[Q-Pochhammer symbol]]
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| | <math>(z;q)_\infty = \prod_{n=0}^\infty (1-zq^n)</math>
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| | Widely used in [[q-analog]] theory. The [[Euler function]] is a special case.
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| |- valign=top
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| | [[Ramanujan theta function]]
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| | <math>\begin{align}
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| f(a,b) &=\sum_{n=-\infty}^\infty a^{\frac{n(n+1)}{2}} b^{\frac{n(n-1)}{2}} \\
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| &= \prod_{n=0}^\infty (1+a^{n+1}b^n)(1+a^nb^{n+1})(1-a^{n+1}b^{n+1})
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| \end{align}</math>
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| | An expression of the [[Jacobi triple product]], also used in the expression of the Jacobi [[theta function]]
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| |- valign=top
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| | [[Riemann zeta function]]
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| | <math>\zeta(z) = \prod_{n=1}^{\infty} \frac{1}{1 - p_n^{-z}}</math>
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| | Here ''p''<sub>''n''</sub> denotes the sequence of [[prime number]]s. This is a special case of the [[Euler product]].
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| |}
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| Note that the last of these is not a product representation of the same sort discussed above, as ζ is not entire. Rather, the above product representation of [[zeta function|ζ(z)]] converges precisely for Re(z) > 1, where it is an analytic function. By techniques of [[analytic continuation]] this function can be extended uniquely to an analytic function (still called ζ(z)) on the whole complex plane except for the point z=1, where it has a simple [[pole_(complex_analysis)|pole]].
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| ==See also==
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| *[[List of trigonometric identities#Infinite product formulae|Infinite products in trigonometry]]
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| *[[Series (mathematics)|Infinite series]]
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| *[[Continued fraction]]
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| *[[Infinite expression (mathematics)|Infinite expression]]
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| *[[Iterated binary operation]]
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| == References ==
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| * {{cite book
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| |last=Knopp
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| |first=Konrad
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| |authorlink=Konrad Knopp
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| |title=Theory and Application of Infinite Series
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| |publisher=[[Dover Publications]]
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| |year=1990
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| |isbn=978-0-486-66165-0
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| |language=English translation
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| }}
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| * {{cite book
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| |last=Rudin
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| |first=Walter
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| |authorlink=Walter Rudin
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| |title=Real and Complex Analysis
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| |edition=3rd
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| |publisher=[[McGraw Hill]]
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| |location=Boston
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| |year=1987
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| |isbn=0-07-054234-1
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| }}
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| * {{Cite book
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| |editor1-last=Abramowitz
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| |editor1-first=Milton
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| |editor1-link=Milton Abramowitz
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| |editor2-last=Stegun
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| |editor2-first=Irene A.
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| |editor2-link=Irene Stegun
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| |title=[[Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]]
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| |publisher=[[Dover Publications]]
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| |year=1972
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| |isbn=978-0-486-61272-0
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| }}
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| ==External links==
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| *[http://mathworld.wolfram.com/InfiniteProduct.html Infinite products from Wolfram Math World]
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| [[Category:Sequences and series]]
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| [[Category:Mathematical analysis]]
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| [[Category:Multiplication]]
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| [[es:Productorio]]
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They have powerful and durable mountain bikes with a high-tech suspension system. Macaskill is tackling the Cuillin Ridge in Scotland, not far from where this two-wheeling genius hails from on the Isle of Skye. They both have some parts that are worse than another's, but it doesn't matter in general. its also important to look at the design of the bike some bikes are designed for specific porpuses that is to say , there bike s designed for sports , transport porpuses while other motobike are designed for prestige , for leisure porpuses before you go the market ldentify your porpuses why you want tobuy the mountai bike or any bike. Certain Haro mountain bike models worth mentioning right here consist of BMX Black trail X1, BMX Haro Partial 16, BMX Haro FIC and so forth.
If you want to get information about some really special and unique forks visit my site. When you have any issues about wherever and tips on how to employ Choosing the right ride for you mountain bike sizing., you can email us with the internet site. "Before breathing exercises it took me 45 minutes to really get into my stride when off road mountain biking. If you don't do a lot of racing on mountain trails then you need a road bike. - Get a bike that is lightweight and made out of carbon fiber materials. UST kits can also improve sturdiness of UST wheels with added sealing capacity in case of any punctures.
The steeper the angles, the more beneficial it would be for stability and high speed pedaling. If you do it as soon as you get home, then its done and you can concentrate on eating and relaxing. Guided by the dolphin trainer, they will perform a series of tricks such as clapping, synchronized jumping, etc. Being humiliated by my buddies was not fun, but there were enough good things about skiing that I wanted to come back for more, just not with my buddies. -Del Plomo Hot Springs: Ideal for bikes, low traffic.
Read product reviews and cycling magazines, research online, and ask for advice at your local bike shop. Cold and long winters at high altitude are perfect conditions for winter recreation. Article Source: mini bikes, minibikes and pocket rockets; At About-minibikes. Next, there is also a seat in the bike for the riders to sit during the riding. Cross country Nearly all of the mountain bikes available could be classed as cross country.
I ran with Hale and we also tried this method, it was effective. We have the best selection of bicycle parts( like wheels, bike handlebars, cycling shoes, bicycle pedals, bike cranks, shifters, bicycle brakes, bicycle tires and much more. Mountain biking amongst the majestic backdrop of the great mountains of Kerala is an unforgettable and thrilling experience. Suspension: Road bikes are built with a sole purpose of providing greater speed; they do not possess this feature, although they have certain materials which absorb the shocks of the uneven roads. They will provide you with a honest price, yet be ready to pay in between $650 and $4,000 on dual suspensions and in between $470 and $670 for hardtails.