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Topic: Infinite series

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  Series (mathematics) - Wikipedia, the free encyclopedia
Series may be finite, or infinite; in the first case they may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way.
Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète.
Asymptotic series, otherwise asymptotic expansions, are not typically convergent infinite series, but sequences of finite approximations each of which is a good asymptotic representation.
en.wikipedia.org /wiki/Infinite_series   (1682 words)

 AllRefer.com - series (Mathematics) - Encyclopedia
An infinite series is a sum of infinitely many terms, e.g., the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ….
Some infinite series converge to a certain value called its limit; i.e., as one adds together progressively more terms, these sums (called the partial sums of the series) form a sequence of values that progressively approach the limit.
A series that does not converge is said to diverge; various tests exist for determining whether or not a given series converges and for determining its limit if it does converge.
reference.allrefer.com /encyclopedia/S/series.html   (330 words)

 Geometric progression - Wikipedia, the free encyclopedia
In mathematics, a geometric progression (also inaccurately known as a geometric series, see below) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence.
An infinite geometric series is an infinite series whose successive terms have a common ratio.
Such a series converges if and only if the absolute value of the common ratio is less than one.
en.wikipedia.org /wiki/Geometric_series   (511 words)

 The Infinite: A Supplement to Aristotle and Mathematics
The infinite series in potentiality by addition is identical with some series of the infinite in potential by division.
The infinite in actuality by addition: With the exception of time, since the only notion of infinite series in potentiality by addition which Aristotle accepts corresponds to a infinite series in potentiality by division, it follows that the only acceptable infinite series in actuality by addition would have to satisfy the same finitistic constraints.
Hence, there is no infinite acutality by addition for sizes or weights, etc. Aristotle's views on infinite time are less clear, but he is committed to some sense of an actual infinite by addition in the case of time (going into the past), but only in a weak sense, since past changes no longer exist.
plato.stanford.edu /entries/aristotle-mathematics/supplement3.html   (1229 words)

 40: Sequences, series, summability
Sequences and series are really just the most common examples of limiting processes; convergence criteria and rates of convergence are as important as finding "the answer".
Taylor series of known functions) are of interest, as well as general methods for computing sums rapidly, or formally.
Series can be estimated with integrals, their stability can be investigated with analysis.
www.math.niu.edu /~rusin/known-math/index/40-XX.html   (588 words)

 Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-10-21)
Infinite Series Involving Pi Date: 12/16/97 at 01:00:59 From: Eric Tak Subject: Infinite series involving pi Sorry, I am just totally confused on this problem.
This series is periodic with period 2.pi and the convention is to take the interval -pi < x < pi.
This means that we can evaluate all the a(i), b(i) and express f(x) as an infinite series in sines and cosines of multiples of x.
mathforum.org /library/drmath/view/56916.html   (543 words)

 infinite series - a Whatis.com definition - see also: infinite sequence
An infinite sequence is a list or string of discrete objects, usually numbers, that can be paired off one-to-one with the set of positive integers {1, 2, 3,...}.
An infinite series is the sum of the values in an infinite sequence of numbers.
Otherwise, the series and its corresponding sequence diverge.
whatis.techtarget.com /definition/0,,sid9_gci804476,00.html   (293 words)

 Selected Problems from the History of the Infinite Series
Infinite Series were used throughout the development of the calculus and it is thus difficult to trace their exact historical path.
Gregory was one of the first to relate trigonometric functions to their infinite series using calculus, although he is primarily only remembered noted for finding the infinite series for the inverse tangent.
That is, given the infinite series for a function, he found a way to calculate the infinite series for the inverse function.
www.math.wpi.edu /IQP/BVCalcHist/calc3.html   (1093 words)

 Series -- Types and Tests (PRIME)
This important series should be thought of as a function in x for all x in the radius of convergence.
A series converges when its sequence of partial sums converges, that is, if the sequence of values given by the first term, then the sum of the first two terms, then the sum of the first three terms, etc., converges as a sequence.
A series is said to converge absolutely if the series still converges when all of the terms of the series are made non-negative (by taking their absolute value).
www.mathacademy.com /pr/prime/articles/serie   (681 words)

 A Radical Approach to Real Analysis
He rather viewed infinite series in a larger context, a context that he makes clear in his article “On divergent series” published in 1760.
On the other hand, as series in analysis arise from the expansion of fractions or irrational quantities or even of transcendentals, it will in turn be permissible in calculation to substitute in place of such series that quantity out of whose development it is produced.”
Euler merely asks that in the case of a series that does not converge, we allow a value determined by the genesis of the series.
www.macalester.edu /aratra/chapt2/chapt2_2a.html   (348 words)

 Could an Infinite Series of Past Events
He argues that sin ce an infinite series of numbers must be "defined" starting with some number, an actually infinite series could not be "formed" by successive addition.
If the series of past events were both infinite and formed by successive addition, then it would be possible to complete a count of all the members of the series <0,–1,..–n,..>.
Even if, in a logical sense, a series of numbers "begins" with zero (since all the other numbers are "defined" in relation to zero), it might still be possible that, in the temporal order of events, an enumeration of all the numbers of the series ends with zero.
stripe.colorado.edu /~morristo/infpast.html   (6005 words)

 Infinite Series
An infinite series is a series which is infinite.
Strictly speaking, the summation can be from zero, one, or other numbers (including negative numbers, up to and including minus infinity), but to qualify as an “infinite” series, there has to be an infinite number of terms.
This infinite series is said to “approach” 2 (as opposed to equal it -- even though it does).
www.halexandria.org /dward018.htm   (303 words)

 Infinite series   (Site not responding. Last check: 2007-10-21)
We extend the concept of a finite series to the situation in which the number of terms increase without bound to give rise to an infinite series.
We explore what is meant by an an infinite series being convergent by considering the partial sums of the series.
We also consider various tests for the convergence of series, in particular we introduce the ratio test, which is applicable to series of positive terms.
www.lboro.ac.uk /research/helm/cal_tryouts/chap16/j16_2.html   (104 words)

 Geometric Series   (Site not responding. Last check: 2007-10-21)
The first thing that must be discussed when working with infinite series is the meaning of convergence of an infinite sequence of real numbers.
An infinite series is formed by adding, successively, the terms of a sequence.
To decide which of the of the corresponding infinite series converge, we need to find the sequence of partial sums.
mathcircle.berkeley.edu /BMC4/Handouts/serie/node2.html   (297 words)

 Time Supplement [Internet Encyclopedia of Philosophy]
There have been serious attempts over the last few decades to construct theories of physics in which spacetime is a product of more basic entities.
An actually infinite entity has its infinitude (that is, its being unending) existing at a specific time, but a potentially infinite entity has its infinitude existing over an interval of time.
The notion of infinite sums of numbers had to be revised so that an infinite series of numbers that decrease sufficiently rapidly can have a finite sum.
www.iep.utm.edu /ancillaries/time-sup.htm   (12230 words)

The idea of an alternating infinite series of plus and minus terms approaching pi is not new.
In the Leibniz series you would have pi to only 3 correct decimal places after seven thousand alternate steps.
If the line AB in Illustration 2 was the distance light travels in one thousand years, the increment subtracted in the last step in the series in Illustration 6 would be near the scale of the subatomic particle known as the quark.
members.aol.com /iterate/Pi.htm   (2066 words)

 [No title]   (Site not responding. Last check: 2007-10-21)
An infinite series is an expression like this:
The figure composed of yellow boxes is a model for the partial sum of a series.
He imagined an arrow flying towards its target and argued that since it would never reach it, that no motion was possible.
www.math.utah.edu /~carlson/teaching/calculus/series.html   (953 words)

 15.1 Infinite Series and Convergence   (Site not responding. Last check: 2007-10-21)
If we have an infinite sequence, we define it to be convergent if, for any positive criterion, q, however small, beyond some term, say the n(q)th, all of the terms are within q of some number, z which we call the limit of the sequence.
Then convergence of the series is defined to be the same as the convergence of that sequence of partial sums.
When a series is absolutely convergent, you can rearrange its terms, differentiate it term by term if terms contain a variable, and perform other manipulations, which may not work for merely convergent series.
www-math.mit.edu /~djk/calculus_beginners/chapter15/section01.html   (292 words)

 Infinite Geometric Series
Suppose that r = 1 then the infinite series is a + a + a + a +...
Suppose that r = -1 then the infinite series is a - a + a - a +...
That is, if you start to add up at the second term of your series instead of the first, then you get an answer of X/2 instead of X. Therefore, the first term of the series was already X/2; so 1/2 = X/2, and X = 1.
mathcentral.uregina.ca /QQ/database/QQ.09.00/carter1.html   (960 words)

 Infinite series (from analysis) --  Encyclopædia Britannica
This particular series is relatively harmless, and its value is precisely 1.
Infinite series are useful in mathematics and in such disciplines as physics, chemistry, biology, and engineering.
Everywhere in a series circuit the current is the same.
www.britannica.com /eb/article-218263?tocId=218263   (923 words)

 Infinite Series
The word "series" in common language implies much the same thing as "sequence", but in mathematics when we talk of a series we are referring to sums of terms in a sequence.
The sums of geometric sequences are called geometric series, and can be shown to converge whenever the ratio of successive terms has magnitude less than 1.
Power series which arise as limiting cases of Taylor Polynomials are called Taylor Series.
www.langara.bc.ca /mathstats/resource/onWeb/calculus/series   (480 words)

 Calculus II (Math 2414) - Series & Sequences - Series - The Basics
The series notation (or summation notation or sigma notation, which ever you prefer) tells us to add all the items from the sequence starting at the value of the index that is below the sigma.  Also note that the letter that we use for the index is not important.  The following two series are identical.
Multiplying infinite series needs to be done in the same manner.  Remember that a series is really a giant summation and so here is what we’re really asking for multiplication,
The basic idea behind index shifts is to start a series at a different value for whatever the reason (and yes, there are legitimate reasons for doing that).
tutorial.math.lamar.edu /AllBrowsers/2414/Series_Basics.asp   (1089 words)

 Notes on the attached paper
The paper sets forth simple methods to decide whether an infinite series or other infinite processes would converge to a rational number or an irrational number.
I have isolated the part of the paper that involves a simple issue of mathematical logic as a self-contained and purely elementary paper carrying the central theorem and its proof.
It is usually very hard to show that an infinite series has an irrational sum.
www.infiniteseriestheorem.org   (1268 words)

 4.1. Series and Convergence   (Site not responding. Last check: 2007-10-21)
Note that while a series is the result of an infinite addition - which we do not yet know how to handle - each partial sum is the sum of finitely many terms only.
Hence, there are different modes of convergence: one mode that applies to series with positive terms, and another mode that applies to series whose terms may be negative and positive.
Then any rearrangement of terms in that series results in a new series that is also absolutely convergent to the same limit.
web01.shu.edu /projects/reals/numser/series.html   (823 words)

 Infinite Facts Series - Home Page 1 at Project HappyChild
The Infinite Facts Series is a collection of interesting facts for kids - ranging from subjects like "The Police Letters alphabet" to Kings & Queens, UK Prime Ministers, Morse Code, the Solar System, Stars & Constellations, Arabic, Roman & Binary Numerals...
*NEW* A long-planned expansion of The Infinite Fact Series has come about with the setting up of "Keywords for Learning", which allows kids to assimilate endless collections of facts through doing "word searches" (these can be printed out for use at home or school).
Facts relating to "Pokémon" and "Harry Potter" [most popular subjects of all for many kids!] can be found in Area 12 and Area 13 at Project HappyChild; there are however already 25 Pokémon wordsearches in the KFL area, from lists sent in by site visitors, and Harry Potter wordsearches are also planned.
www.happychild.org.uk /home1.htm   (340 words)

 A Radical Approach to Real Analysis
These simple facts do not always hold for infinite sums.
It takes a little more effort to see that rearrangements are not always allowed, but the effort is rewarded in the observation that some very strange things are happening here.
Click here to explore partial sums of this and other rearrangements of the alternating harmonic series.
www.macalester.edu /aratra/chapt2/chapt2_1b.html   (223 words)

 Sum of Infinite Series
A series that converges has a finite limit, that is a number that is approached. 
can be written as an infinite series.  Write it as a series and tell if it diverges/converges.  If it converges, find the sum.
It's now an infinite series.  Mickey spots the common ratio of the series as.0027/.27 =.01.  Therefore, it converges!!  He use the formula and presto:
www.geocities.com /CapeCanaveral/Launchpad/2426/page135.html   (449 words)

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