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Topic: Geometric series

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	PlanetMath: geometric series
		A geometric series is a series of the form
		The partial sums of a geometric series are given by
		This is version 12 of geometric series, born on 2002-01-03, modified 2006-10-25.
	planetmath.org /encyclopedia/GeometricSeries.html (104 words)

	Geometric Series
		All of these forms are equivalent, and the formulation above may be derived from polynomial long division.
		As you can see in the screen-capture to the right, entering the values in fractional form and using the "convert to fraction" command still results in just a decimal approximation to the answer.
		Using the summation formula to show that the geometric series "expansion" of
	www.purplemath.com /modules/series5.htm (422 words)

	PlanetMath: geometric series
		A geometric series is a series of the form
		The partial sums of a geometric series are given by
		This is version 12 of geometric series, born on 2002-01-03, modified 2006-10-25.
	www.planetmath.org /encyclopedia/InfiniteGeometricSeries.html (104 words)

	Geometric progression Summary
		A geometric series is the sum of the terms of a geometric sequence.
		Geometric series may be finite, such as the series in the preceding sentence, or infinite, such as 1+2+4+..., where the three dots indicate that the series follows this pattern forever.
		In mathematics, a geometric progression (also known as a geometric sequence, and, inaccurately, 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.
	www.bookrags.com /Geometric_progression (1158 words)

	Kids.Net.Au - Encyclopedia > Geometric series
		A geometric series is a sum of terms in which two successive terms always have the same ratio.
		It is called a geometric series because it occurs when comparing the length, area, volume, etc. of a shape in different dimensions.
		An infinite geometric series is an infinite series whose successive terms have a common ratio.
	www.kids.net.au /encyclopedia-wiki/ge/Geometric_series (305 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.
		Assume that a is positive and the sum of the finite geometric series is.
		Suppose that r = 1 then the infinite series is a + a + a + a +...
	mathcentral.uregina.ca /QQ/database/QQ.09.00/carter1.html (949 words)

	.999999... = 1?
		Series are introduced and studied rigorously in Calculus, where a distinction is made: some series are convergent, some are divergent.
		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)

	Geometric Series
		Thus a geometric series of cash flows is one in which the rate of increase or decrease of cash flows occurs at a faster rate than for a simple linear gradient or arithmetic series.
		An example of a geometric series would be an account that is opened with some initial deposit and in which all subsequent deposits increase at a fixed percentage rate.
		In problems involving geometric gradient series the constant percentage rate of change of cash flow is denoted by g; the interest rate is denoted as usual by i.
	coen.boisestate.edu /mkhanal/geometri.htm (534 words)

	BBC Education - AS Guru - Maths - Pure - Sequences and Series - Geometric Series
		Geometric series are sequences in which successive terms are in the same ratio.
		Geometric series, whilst not named as such by Euclid, were referred to as numbers in continued proportion.
		The lengths of string or tube that produce twelve equally spaced notes form a geometric series starting at the original length and ending with half that length.
	www.bbc.co.uk /education/asguru/maths/13pure/03sequences/18geometric/index.shtml (526 words)

	MathComplete.com - Sequence and Series - Tutorial (Site not responding. Last check: 2007-11-05)
		A series is formed by the sum of the terms of a sequence.
		Arithmetic series is a sum of a number each of which, after the first, is obtained by adding to the preceding number a constant number called the common difference.
		Geometric series is a sum of a number each of which, after the first, is obtained by multiplying the preceding number by a constant number called common ratio.
	www.mathcomplete.com /tutorial/sequence/default.asp?pg=2 (271 words)

	Geometric Series
		Although the algebra for geometric series of negative numbers is the same as that for positive numbers I include it as a special case because all of us, even full-fledged mathematicians, are more reluctant to use negative numbers than positive ones.
		As should become clear, a geometric series is well suited for assigning values in such a way that each jump lowers or keeps same the value of a position.
		This value of 1 is the initial value of a geometric series extending to the left and of another geometric series extending to the right.
	www.geocities.com /Athens/Delphi/5136/geoser2.html (2816 words)

	Geometric Series (Site not responding. Last check: 2007-11-05)
		Geometric Sequences are those sequences in which each term is obtained by multiplying the term before it by the same number.
		A series is the sum of a sequence.
		A geometric series will converge if the common ration r is a proper fraction.
	www.mecca.org /~halfacre/MATH/plesson24.htm (184 words)

	[No title]
		A geometric series is a series which follows the pattern, EMBED "Equation" \* mergeformat where EMBED "Equation" \* mergeformat is the initial term and EMBED "Equation" \* mergeformat is a ratio (i.e., EMBED Equation.DSMT4 , EMBED Equation.DSMT4 , EMBED Equation.DSMT4 , etc.).
		SERIES WITH BOTH POSITIVE AND NEGATIVE TERMS The next sets of series tests are those that apply to series with both negative and positive terms (known as alternating series).
		An alternating series is a series of the form EMBED "Equation" "Word Object6" \* mergeformat , in other words, a series whose terms alternate between positive and negative.
	www.utexas.edu /student/utlc/lrnres/mstc/a315/calchandouts/infiniteseries.doc (2062 words)

	7.3 - Geometric Sequences
		A geometric sequence is a sequence in which the ratio consecutive terms is constant.
		An infinite geometric series is the sum of an infinite geometric sequence.
		The magnitude of the ratio can't equal one because that the series wouldn't be geometric and the sum formula would have division by zero.
	www.richland.edu /james/lecture/m116/sequences/geometric.html (707 words)

	Geometric Sequences and Series (Site not responding. Last check: 2007-11-05)
		Since all geometric sequences have the same basic form, we can write a general formula (or function) which can be used to describe all geometric sequences.
		Since the absolute value of each subsequent term of an geometric sequence is often increasing, the only geometric sums that we can always calculate are finite geometric sums.
		Since the number at the top of the sigma or summation symbol is an infinity, we need to do two thing: determine if this series is geometric and if it is, determine if the common ratio has an absolute value less than 1.
	fym.asu.edu /~fym/mat117-internet/sequences_and_series_notes/geometric-sequences-and-series/Geometric_Sequences_and_Series.html (1712 words)

	Convergence of Series
		Whereas the harmonic series does not converge, the alternating harmonic series does converge due to the alternation of signs.
		In other words, the series we are interested in eventually becomes smaller than a geometric series which converges.
		This is interesting because, in addition to showing that the series converges, it shows that it converges to a number smaller than 3.
	www.ugrad.math.ubc.ca /coursedoc/math101/notes/series/convergence.html (579 words)

	SparkNotes: Sequences and Series: Geometric Sequences
		A geometric sequence is a sequence in which the ratio of any term to the previous term is constant.
		The sum of a finite geometric sequence (the value of a geometric series) can be found according to a simple formula.
		Deciding whether an infinite geometric series is convergent or divergent, and finding the limits of infinite geometric series are only two of many topics covered in the study of infinite geometric series.
	www.sparknotes.com /math/precalc/sequencesandseries/section3.rhtml (222 words)

	Geometric Series (Site not responding. Last check: 2007-11-05)
		Summation notation is explained in conjunction with arithmetic series.
		Summation notation can be used with geometric sequences or any sequence that can be expressed in explicit form.
		For geometric series you do not have to know the nth term which means that not as much work is required for finding sums of geometric series.
	www.algebralab.org /lessons/lesson.aspx?file=Algebra_GeoSeries.xml (371 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.
		This looks like a geometric series but won't actually be one unless the coefficients are all equal (or are themselves a geometric sequence).
	www.langara.bc.ca /mathstats/resource/onWeb/calculus/series/index.htm (480 words)

	Geometric Series (Site not responding. Last check: 2007-11-05)
		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.
		The limit of the sequence of partial sums is said to be the sum of the series.
	mathcircle.berkeley.edu /BMC4/Handouts/serie/node2.html (297 words)

	SparkNotes: Sequences and Series: Terms and Formulae
		Divergent Series - A series whose limit as n→∞ is either ∞ or - ∞.
		Finite Series - A series which is defined only for positive integers less than or equal to a certain given integer.
		Geometric Sequence - A sequence in which the ratio between each term and the previous term is a constant ratio.
	www.sparknotes.com /math/precalc/sequencesandseries/terms.html (388 words)

	Geometric Series (Site not responding. Last check: 2007-11-05)
		Another common type of series is the geometric series (also called a geometric progression).
		Again to specify a geometric series uniquely we need to know the first term as well as the common ratio.
		A geometric series is uniquely specified by the values of a and r.
	www.ucl.ac.uk /Mathematics/geomath/level2/series/ser3.html (176 words)

	The Lydian-Milesian Geometric Electrum (Ancient Coins of Miletos)
		They are often referred to as “geometric” issues and the obverse design is sometimes called a “collapsing square” in the literature.
		They have been described as the first coins ever made with both an obverse and a reverse type (as opposed to a simple incuse reverse), and the degree of wear many of them exhibit suggests that they were long in circulation.
		Obverse: “geometric pattern.” Reverse: “incuse square consisting of seven spokes, striations, and central dot.” The reverse is noted to be similar to Rosen 285, also an EL twenty-fourth (0.593 g) with seven spokes emanating from a central dot, but in this case the obverse is a lion’s paw seen from above.
	rjohara.net /coins/geometric-electrum (629 words)

	Geometric Series by Brandon Rice and Sarah Sticher (Site not responding. Last check: 2007-11-05)
		Geometric Series by Brandon Rice and Sarah Sticher
		A geometric series is the sumation of a geometric progression.
		The series will be consisting of terms that get smaller and smaller and will converge on a specific value.
	coweb.math.gatech.edu:8888 /calculus/3173 (193 words)

	Infinite Geometric Series
		We learned that a geometric series has the form
		Recall that a rational number in decimal form is defined as a number such that the digits repeat.
		We can use a geometric series to find the fraction that corresponds to a repeating decimal.
	www.ltcconline.net /greenl/courses/103b/seqSeries/INFGEO.HTM (200 words)

	[No title]
		The growth of trade and commerce, particularly lending and borrowing demanded an understanding of both simple and compound interest which probably stimulated the interest in arithmetic and geometric series.
		The industry is gradually coming to the deadlock with the capital expenditures rising at geometric series (just recall the cost of modern factories), which is affordable to fewer players, while the results of all those multi-billion expenditures are increasingly less impressive.
		The harmonic series is far less widely known than the arithmetic and geometric series.
	www.lycos.com /info/geometric-series.html (548 words)

	INF
		Oresme's proof of the divergence of the harmonic series was lost, and later reproved by Johan Bernoulli.
		Adding up the first n terms of a series is what is referred to as the nth partial sum, and is denoted as Sn.
		Geometric series are a good starting point for any introduction to infinite series.
	isolatium.uhh.hawaii.edu /m206L/lab11/inf/inf.htm (668 words)

	Hotmath Solution Finder
		Determine whether the sequence is arithmetic, geometric, or neither.
		Calculate the sum of the six terms of the geometric series.
		Calculate the sum of the infinite geometric series.
	www.hotmath.com /help/gt/genericalg2/section_9_3.html (395 words)

	Leaving Cert. Higher Level Maths - Sequences And Series - Geometric Sequences
		Geometric sequences are sequences formed when the preceding term of a sequence is multiplied by a constant called the common ratio.
		So, in order to calculate any terms in a geometric series we have to have the common ratio, r, or other terms of the sequence (i.e.
		Geometric Series are formed when a geometric sequence is summed.
	www.netsoc.tcd.ie /~jgilbert/maths_site/applets/sequences_and_series/geometric_sequences.html (468 words)

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