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# Topic: Infinite sums

 P-adic number A definite meaning is given to these sums based on Cauchy sequences using the p-adic metric[?]. Intuitively, as opposed to p-adic expansions which extend to the right as sums of ever smaller, increasingly negative powers of the base p (as is done for the real numbers as described above), these are numbers whose p-adic expansion to the left are allowed to go on forever. The partial sums of this latter series are the elements of the given series. www.ebroadcast.com.au /lookup/encyclopedia/p-/P-adic_number.html   (1234 words)

 Nirenberg: Paradoxes of Infinity Therefore, the sum of the infinite series, in other words, the limit of these sums, is just 2. But adding infinite series is not an easy task (unless your series happens to be easy), and even now we don't know how to add a lot of series that appear quite natural. Infinite series are not merely a mathematical curiosity: they play very important roles in scientific theory and practice. nyjm.albany.edu:8000 /~rn774/spring97/infinity.html   (2906 words)

 Info Node: (calc.info)Summations   (Site not responding. Last check: 2007-10-11) Also, of course, sums in which the lower and upper limits are both numbers can always be evaluated just by grinding them out, although Calc will use closed forms whenever it can for the sake of efficiency. For example, `sum(prime(k)*k^2, k, 1, 20)' is the sum of the squares of all prime numbers from 1 to 20; the `prime' predicate returns 1 if its argument is prime and 0 otherwise. You can read this expression as "the sum of `k^2', where `k' is prime." Indeed, `sum(prime(k)*k^2, k)' would represent the sum of *all* primes squared, since the limits default to plus and minus infinity, but there are no such sums that Calc's built-in rules can do in closed form. www.math.psu.edu /bin/info2www?(calc.info)Summations   (1252 words)

 Summation of S(N)? However, if S(N) has infinite cardinality (as is generally supposed), then it follows that the process of summation of all of the members of S(N) (summation of an infinite number of finite natural numbers, all but one of which is greater than or equal to 1) must produce an infinite result. In fact one can show that the sum of an infinite number of naturals is a natural exactly when only a *finite* number of the summands are non-zero (assuming zero to be in our naturals). We can, as part of calculus, define an infinite sum in terms of limits IF the limit exists which turns out to be true only if the terms go to 0. www.physicsforums.com /showthread.php?t=66277   (1257 words)

 L.d.a.s in Cardinality Table Since these infinite sums may not play a convincing role in any rigorous mathematical proof of the complete coverage of the integers, it seems important to pursue this line of argument by showing how the coverage of the odd integers increases in finite sums as the size (hence the number) of l.d.a.s employed is increased. The first summation which might come to mind is the finite sum counterpart of the infinite sum used earlier to indicate that all the odd integers are covered in the abstract predecessor tree. This is consistent with the notion that 2 s-steps roughly equal 1 b-step and suggests that the triangle summed to determine the integer coverage should have twice the span in the a axis that it has in the b axis. www-personal.ksu.edu /~kconrow/ldaincar.html   (1307 words)

 Springer Online Reference Works A sequence of elements (called the terms of the given series) of some linear topological space and a certain infinite set of their partial sums (called the partial sums of the series) for which the notion of a limit is defined. A condition for the convergence of a series which does not use the notion of its sum is the Cauchy criterion for the convergence of a series. Already the scientists of Ancient Greece had arrived at the notion of infinite sums: the sum of the terms of an infinite geometric progression with a positive ratio less than 1 can be found in their studies. eom.springer.de /s/s084670.htm   (2297 words)

 Series (mathematics) - Wikipedia, the free encyclopedia The idea of an infinite series expansion of a function was first conceived in India by Madhava in the 14th century, who also developed the concepts of the power series, the Taylor series, the Maclaurin series, rational approximations of infinite series, and infinite continued fractions. He discovered a number of infinite series, including the Taylor series of the trigonometric functions of sine, cosine, tangent and arctangent, the Taylor series approximations of the sine and cosine functions, and the power series of the radius, diameter, circumference, angle θ, π and π/4. In Europe however, the investigation of the validity of infinite series is considered to begin with Gauss in the 19th century. en.wikipedia.org /wiki/Infinite_series   (2673 words)

 Hilbert's Paradox and 0=1 As for the sum, so as 0=1; the problem with that is that an alternating series converges to diffent sums depending on how the terms are grouped, as you have shown. The reason addition is not associative in the infinite case is because defining it to be associative in an infinite case is because it cannot be done without rendering numbers useless. But infinite isn't a number so we can't judge if it is even or not, meaning that the grouping of these terms is inconclusive. www.physicsforums.com /showthread.php?t=31471   (2358 words)

 GNU Emacs Calc 2.02 Manual The formula is taken from the top of the stack; the command prompts for the name of the summation index variable, the lower limit of the sum (any formula), and the upper limit of the sum. Sums of trigonometric and hyperbolic functions are transformed to sums of exponentials and then done in closed form. As a special feature, if the limits are infinite (or omitted, as described above) but the formula includes vectors subscripted by expressions that involve the iteration variable, Calc narrows the limits to include only the range of integers which result in legal subscripts for the vector. www.delorie.com /gnu/docs/calc/calc_274.html   (931 words)

 Infinity and Probability Two sets are equivalent or, which is the same, have the same number of elements when there exists a 1-1 correspondence between their elements. What sets the finite and infinite sets apart is that infinite sets are equivalent to their own subsets which may never happen with finite sets however large. Infinite products are as legitimate objects of math study as are infinite sums - series. www.cut-the-knot.org /Probability/infinity.shtml   (1742 words)

 .999999... = 1? All infinite decimal fractions, like 0.999..., are shown to correspond to convergent series (which converge to their respective sums.) 0.999... The sum of this geometric series is known to be 100(4/9)/(1 - 4/9) = 80. However, the action came out in the form of an infinite series, and summation of that series was virtually impossible in the absence of the geometrical point of view and the invariance principle. www.cut-the-knot.org /arithmetic/999999.shtml   (2124 words)

 PlanetMath: series In a context where this distinction does not matter much (this is usually the case) one identifies a series with its sum, if the latter exists. Traditionally, as above, series are infinite sums of real numbers. So in full generality the terms could be complex numbers or even elements of certain rings, fields, and vector spaces. planetmath.org /encyclopedia/Series.html   (145 words)

 Analysis, Convergence, Series, Complex Analysis - Numericana Infinite sums may sometimes be evaluated with Fourier Series. A double sum is often the product of two sums, which may be Fourier series. At a jump, the sum of a Fourier series is the half-sum of its left and right limits. home.att.net /~numericana/answer/analysis.htm   (4095 words)

 Poster Project, What Is Scientific Truth Poster, Zeno's Paradoxes   (Site not responding. Last check: 2007-10-11) An infinite sum is a sum which has an infinite number of summands. For example, in the equation 2+3=5, the summands are 2 and 3.) Infinite sums are closely allied to notions of sequences and their limits. In fact, infinite sums can be thought of as the limit of a particular sequence: the sequence whose nth term is the sum of the first n summands of the sum. www.math.sunysb.edu /posterproject/www1/materials/truth/theorem2.html   (955 words)

 On Feinstein's Collatz is Unprovable Proof At every stage in the development of the abstract tree, a node represents an infinite set of elements in some {c[d]} which must include members in {0[3]}, {1[3]}, and {2[3]} with equal density. In my web page on "counting", I go through a number of different infinite summations of the densities of the integers in various infinite sets in one or two dimensions (and, in other pages, a bunch of finite summations) to gain/give an impression of how (rapidly) the paths reach leaf nodes in the l.d.a.s. The density of the even integers in the infinite series of powers of 2 multiples of the odd numbers in the predecessor tree have also been summed, and prove to have the identical overall density that the odd numbers have. www-personal.ksu.edu /~kconrow/cafproof.html   (1156 words)

 SummationNotation - PineWiki This can be done using a sum over all indices that are members of a given index set, or in the most general form satisfy some given predicate (with the usual set-theoretic caveat that the objects that satisfy the predicate must form a set). Such a sum is written by replacing the lower and upper limits with a single subscript that gives the predicate that the indices must obey. Without the rule that the sum of an empty set was 0 and the product 1, we'd have to put in a special case for when one or both of A and B were empty. pine.cs.yale.edu /pinewiki/SummationNotation   (1686 words)

 Nicholas of Cusa and the Infinite The former is a view from the finite upward toward the unattainable and incomprehensible infinite, while the latter is an incomprehensible view from the infinite downward toward the finite that is identical with the infinite. Thus, the actual infinite ultimately had to be explicitly affirmed in mathematics in order to provide a foundation for the numbers used in both analytic geometry and calculus. The history of the Infinite thus reveals in both mathematics and philosophy a development of increasingly subtle thought in the form of a dialectical dance around the ineffable and incomprehensible Infinite. www.integralscience.org /cusa.html   (5490 words)

 Infinite product - Wikipedia, the free encyclopedia The value zero is treated specially in order to get results analogous to those for infinite sums. This allows the translation of convergence criteria for infinite sums into convergence criteria for infinite products. One important result concerning infinite products is that every entire function f(z) (i.e., every function that is holomorphic over the entire complex plane) can be factored into an infinite product of entire functions each with at most a single zero. en.wikipedia.org /wiki/Infinite_product   (436 words)

 Infinite Geometric Series You have probably already seen the exprression for the sum of a finite geometric series but I want to develope it just to make sure we are using the same notation. Suppose that r = 1 then the infinite series is a + a + a + a +... Since the sum does not approach a specific value as n increases, we say that the sum of the infinite geometric series does not exist. mathcentral.uregina.ca /QQ/database/QQ.09.00/carter1.html   (949 words)

 Zeno's Paradoxes (Stanford Encyclopedia of Philosophy) One central element of the this theory of the ‘transfinite numbers’ is a precise definition of when two infinite collections are the same size, and when one is bigger than the other — with such a definition in hand it is then possible to order the infinite numbers just as the finite numbers are ordered. He might have had the intuition that any infinite sum of finite quantities, since it grows endlessly with each new term must be infinite, but one might also take this kind of example as showing that some infinite sums are after all finite. Of course, one could again claim that some infinite sums in fact have finite totals, and in particular that the sum of these pieces is 1 × the total time, which is of course finite (and again a complete solution would demand a rigorous account of infinite summation, like Cauchy's). plato.stanford.edu /entries/paradox-zeno   (10017 words)

 Derivatives and Integrals   (Site not responding. Last check: 2007-10-11) Many properties of continuous bodies depend upon weighted sums, which to be exact must be infinite weighted sums - a problem tailor-made for the integral. For example, finding the center of mass of a continuous body involves weighting each element of mass by its distance from an axis of rotation, a process for which the integral is necessary if you are going to get a precise value. A vast number of physical problems involve such infinite sums in their solutions, making the integral an essential tool for the physical scientist. hyperphysics.phy-astr.gsu.edu /hbase/math/derint.html   (362 words)

 phil 105 fall lecture notes part 1 on zeno Zeno assumes 'for the sake of argument' that motion occurs, and shows that it follows that an infinite number of distances must be covered in a finite time – something he believes impossible. Olber's: "Space is infinite, and stars evenly distributed on average, so there is a star in every direction, so why isn’t the night sky light?" – a surprising and false conclusion. The partial sums grow without end but they don't grow indefinitely large; they approach ever closer to 1 without exceeding it - 1 is the 'limit' of the sequence. www.uic.edu /classes/phil/phil105nh/105lectures/105lecture03.html   (638 words)

 infinite - Definitions from Dictionary.com Usage Note: Infinite is sometimes grouped with absolute terms such as unique, absolute, and omnipotent, since in its strict mathematical sense infiniteness is an absolute property; some infinite sets are smaller than others, but they are no less infinite. In nontechnical usage, of course, infinite is often used to refer to an unimaginably large degree or amount, and in these cases it is acceptable to modify or compare the word: Nothing could give me more infinite pleasure than to see you win. Used very loosely as in: "This program producesinfinite garbage." "He is an infinite loser." The word most likelyto follow `infinite', though, is hair. dictionary.reference.com /browse/infinite   (663 words)

 APPROXIMATIONS OF SUMS OF INFINITE SERIES In Section 8.2 of the text the sum of an infinite series is defined as is the partial sum of the first n terms of the series. Sometimes it may be advantageous to use Sum and value instead of just sum. www.math.wpi.edu /Course_Materials/MA1023C98/infinite/node1.html   (498 words)

 Supertasks (Stanford Encyclopedia of Philosophy) To this the latter might reply that the assertion that the sum of the series is 1 presupposes no infinite sum, since, by definition, the sum of a series is the limit to which its partial (and so finite) sums approach. The reason is that a property shared by the partial sums of a series does not have to be shared by the limit to which those partial sums tend. Davies [2001] has proposed a model of an infinite machine (an infinite machine is a computer which can carry out an infinite number of computations within a finite length of time) based on the Newtonian dynamics of continuous media which reveals the nature of the difficulty. plato.stanford.edu /entries/spacetime-supertasks   (10970 words)

 Question Corner -- Numbers Defined By Infinite Sums The second sum is often called the harmonic series and is known to diverge. Continuing in this way, we see that the sum eventually passes every finite number, so the total sum is infinite. There are infinitely many real numbers (in fact, what mathematicians call "uncountably infinitely many" which is even more than just "infinitely many"!), so there are plenty of numbers that nobody has had occasion to think about in a special way. www.math.toronto.edu /mathnet/questionCorner/specialnumber.html   (487 words)

 What's the sum of all integers? | Ask MetaFilter Roughly speaking, that means the sum of the first N terms in the series grows closer and closer to a fixed value as N becomes larger. To put it another way, the sum exceeds 1-x, where x is any positive number, no matter how small (but remember that "positive" implies "non-zero"), in a finite number of terms of the series. The sum of all integers is the highest integer, multiplied by that integer plus 1, divided by 2. ask.metafilter.com /mefi/25060   (2671 words)

 Generalized Matrix This is simply matrix multiplication, but the matrix is infinite, and the sums are infinite too. The linear operator t is hermitian iff its infinite matrix m is hermitian. Integrals are additive, so the integral of (f+g)h is the sum of the integral of fh plus the integral of gh. www.mathreference.com /la-xf,gm.html   (1337 words)

 Operating on Sequences   (Site not responding. Last check: 2007-10-11) This is an infinite sum, and f is defined wherever it converges. Since t() is linear across infinite sums, the resulting function is called a taylor series, rather than a taylor sequence. To work properly, the inverse function is linear with respect to infinite sums. www.mathreference.com /la-xf,sum.html   (459 words)

 Math1b, Summer 2005, Introduction to Functions and Calculus II More important is the ability to apply the integration as appropriate in problem solving; we will devote time to developing your skill in doing this. In the second unit of the course we will study infinite sums. For this reason the process of slicing, approximating a quantity on a slice, summing over all the slices to get a Riemann Sum, and taking the appropriate limit is the real heart of the applications section. www.math.harvard.edu /archive/1b_summer_05/syllabus.html   (1771 words)

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