|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{merge|Geometric progression|discuss=Talk:Geometric series#Merger proposal|date=November 2013}}
| | In the quest to get rid of weight, there are a range of foods which are not just healthy plus nutritious, however help your body lose weight naturally plus grow a bodys rate of fat reduction. Incorporating some or all of these foods into your daily diet will accelerate fat reduction, build a healthier body, and place we found on the road to a healthier lifestyle.<br><br>Raspberry ketone appears to aid with fat in the liver plus visceral adipose (adnominal fat). It also significantly improved norepinephrine-induced lipolysis (the decomposition of fat) inside some rat fat cells. Keep in mind that biomedical research is constantly completed in a lab on rats initial but it will or may not have the same impact whenever employed on people. It is just rather promising preliminary research. Scientists speculate that raspberry ketone stimulates the energy metabolism via a system similar to which of capsaicin. Capsaicin, a compound found in chili peppers, has been looked at in humans for weight reduction. Both capsaicin plus raspberry ketone are being studied as a catalyst inside thermogenisis the heating plus burning of fat in the body.<br><br>When Dr. Mehmet Oz described raspberry ketone as the "magic fat burner in a bottle" on his talk show, he established a fat-burning phenomenon. Learn about the research showing how it can help you shed pounds more easily. And read about what to find and how much to take if you do choose to invest your bucks inside this fat reduction supplement.<br><br>Additionally a ski helmet can protect your head within the snow and the melted cold water. At freezing conditions this protection may save the life! And and how rock hard these ski helmets are against collisions!<br><br>Dr. Oz views [http://safedietplansforwomen.com/raspberry-ketones raspberry ketone] as his "number one weight loss miracle in a bottle," he declared on his show recently. This compound, which is made from red raspberries, helps to control adiponectin, which is a hormone that stimulates your body to boost your metabolism. Some say it also suppresses their appetite. The result: your body burns fat more effectively and faster. Think you can just substitute red raspberries?<br><br>Though there is a lot of glucose in the blood the lack of insulin signifies it cannot reach the cells where it is actually raspberry ketone diet required for energy. The cells then send emergency signals plus the bodies response is to break down fat shops as substitution vitality. As fat is broken down, poisonous acidic chemicals, called ketones, are released into the bloodstream.<br><br>Would we like to burn more fat easier? Along with the healthy diet and exercise program? Don't go out and begin eating a ton of raspberries from the marketplace. You would have to consume 9- pounds to get the same benefit because 1 tiny liquid dosage of raspberry ketone. Raspberry ketone is the main substance found in red raspberries. It appears to regulate adiponectin, a protein inside the body which is selected to control metabolism. Adiponectin helps the body burn fat quicker. Higher shops of adiponectin inside the body have been associated with fewer fat shops. Dr. Oz recommends 100mg a day for an powerful supplement to your healthy diet plus exercise plan. Typical bottles of the unique supplement cost between $12.00 and $22.00 on average.<br><br>Raspberry Ketone: Since Dr. Oz announced it his #1 diet miracle in a bottle, raspberry ketones has skyrocketed in demand. But, ConsumerLab cautions which though it's supposed to aid with fat reduction by marketing fat breakdown, no safety research have been conducted at amounts inside supplements. In addition, the analysis was performed on mice - not people. |
| {{insufficient inline citations|date=July 2013}}
| |
| {{about|infinite geometric series|finite sums|geometric progression}}
| |
| [[File:GeometricSquares.svg|thumb|right|Each of the purple squares has 1/4 of the area of the next larger square (1/2×{{nowrap|1/2}} = 1/4, 1/4×1/4 = 1/16, etc.). The sum of the areas of the purple squares is one third of the area of the large square.]]
| |
| {{Calculus |Series}}
| |
| | |
| In [[mathematics]], a '''geometric series''' is a [[series (mathematics)|series]] with a constant ratio between successive [[term (mathematics)|terms]]. For example, the series
| |
| | |
| :[[1/2 + 1/4 + 1/8 + 1/16 + · · ·|<math>\frac{1}{2} \,+\, \frac{1}{4} \,+\, \frac{1}{8} \,+\, \frac{1}{16} \,+\, \cdots</math>]]
| |
| | |
| is geometric, because each successive term can be obtained by multiplying the previous term by {{sfrac|2}}.
| |
| | |
| Geometric series are one of the simplest examples of [[infinite series]] with finite sums, although not all of them have this property. Historically, geometric series played an important role in the early development of [[calculus]], and they continue to be central in the study of [[convergent series|convergence]] of series. Geometric series are used throughout mathematics, and they have important applications in [[physics]], [[engineering]], [[biology]], [[economics]], [[computer science]], [[queueing theory]], and [[finance]].
| |
| | |
| ==Common ratio==
| |
| The terms of a geometric series form a [[geometric progression]], meaning that the ratio of successive terms in the series is constant. This relationship allows for the representation of a geometric series using only two terms, ''r'' and ''a''. The term ''r'' is the common ratio, and ''a'' is the first term of the series. As an example the geometric series given in the introduction,
| |
| | |
| :[[1/2 + 1/4 + 1/8 + 1/16 + · · ·|<math>\frac{1}{2} \,+\, \frac{1}{4} \,+\, \frac{1}{8} \,+\, \frac{1}{16} \,+\, \cdots</math>]]
| |
| | |
| may simply be written as
| |
| | |
| :<math> a + a r + a r^2 + a r^3 \cdots </math> , with <math> r = \frac{1}{2} </math> and <math> a = \frac{1}{2} </math> .
| |
| | |
| The following table shows several geometric series with different common ratios:
| |
| | |
| {| class="wikitable" style="margin: 1em auto 1em auto"
| |
| |-
| |
| ! Common ratio, ''r''
| |
| ! Start term, ''a''
| |
| ! Example series
| |
| |-
| |
| | style="text-align:center;"| 10
| |
| | style="text-align:center;"| 4
| |
| | 4 + 40 + 400 + 4000 + 40,000 + ···
| |
| |-
| |
| | style="text-align:center;"| 1/3
| |
| | style="text-align:center;"| 9
| |
| | 9 + 3 + 1 + 1/3 + 1/9 + ···
| |
| |-
| |
| | style="text-align:center;"| 1/10
| |
| | style="text-align:center;"| 7
| |
| | 7 + 0.7 + 0.07 + 0.007 + 0.0007 + ···
| |
| |-
| |
| | style="text-align:center;"| 1
| |
| | style="text-align:center;"| 3
| |
| | 3 + 3 + 3 + 3 + 3 + ···
| |
| |-
| |
| | style="text-align:center;"| −1/2
| |
| | style="text-align:center;"| 1
| |
| | 1 − 1/2 + 1/4 − 1/8 + 1/16 − 1/32 + ···
| |
| |-
| |
| | style="text-align:center;"| –1
| |
| | style="text-align:center;"| 3
| |
| | 3 − 3 + 3 − 3 + 3 − ···
| |
| |}
| |
| | |
| The behavior of the terms depends on the common ratio ''r'':
| |
| :If ''r'' is between −1 and +1, the terms of the series become smaller and smaller, approaching zero in the limit and the series converges to a sum. In the case above, where ''r'' is one half, the series has the sum one.
| |
| :If ''r'' is '''greater than one''' or '''less than minus one''' the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, and the series has no sum. (The series [[Divergent series|diverges]].)
| |
| :If ''r'' is '''equal to one''', all of the terms of the series are the same. The series diverges.
| |
| :If ''r'' is '''minus one''' the terms take two values alternately (e.g. 2, −2, 2, −2, 2,... ). The sum of the terms [[Oscillation (mathematics)|oscillates]] between two values (e.g. 2, 0, 2, 0, 2,... ). This is a different type of divergence and again the series has no sum. See for example [[Grandi's series]]: 1 − 1 + 1 − 1 + ···.
| |
| | |
| {{anchor|Sum}}
| |
| ==Sum==
| |
| The [[sum]] of a geometric series is finite as long as the absolute value of the ratio is less then 1; as the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series being infinite. The sum can be computed using the [[self-similarity]] of the series.
| |
| | |
| ===Example===
| |
| [[File:GeometricCircles.png|300px|thumb|right|A [[self-similar]] illustration of the sum ''s''. Removing the largest circle results in a similar figure of 2/3 the original size.]]
| |
| Consider the sum of the following geometric series:
| |
| :<math>s \;=\; 1 \,+\, \frac{2}{3} \,+\, \frac{4}{9} \,+\, \frac{8}{27} \,+\, \cdots</math>
| |
| This series has common ratio 2/3. If we multiply through by this common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on:
| |
| :<math>\frac{2}{3}s \;=\; \frac{2}{3} \,+\, \frac{4}{9} \,+\, \frac{8}{27} \,+\, \frac{16}{81} \,+\, \cdots</math>
| |
| This new series is the same as the original, except that the first term is missing. Subtracting the new series (2/3)''s'' from the original series ''s'' cancels every term in the original but the first:
| |
| :<math>s \,-\, \frac{2}{3}s \;=\; 1,\;\;\;\mbox{so }s=3.</math>
| |
| A similar technique can be used to evaluate any [[self-similar]] expression.
| |
| | |
| ===Formula===
| |
| | |
| For <math>r\neq 1</math>, the [[Finite geometric series|sum of the first ''n'' terms of a geometric series]] is:
| |
| | |
| :<math>a + ar + a r^2 + a r^3 + \cdots + a r^{n-1} = \sum_{k=0}^{n-1} ar^k= a \, \frac{1-r^{n}}{1-r},</math>
| |
| | |
| where ''a'' is the first term of the series, and ''r'' is the common ratio. We can derive this formula as follows:
| |
| :<math>
| |
| \begin{align}
| |
| &\text{Let }s = a + ar + ar^2 + ar^3 + \cdots + ar^{n-1}. \\[4pt]
| |
| &\text{Then }rs = ar + ar^2 + ar^3 + ar^4 + \cdots + ar^{n} \\[4pt]
| |
| &\text{Then }s - rs = a-ar^{n} \\[4pt]
| |
| &\text{Then }s(1-r) = a(1-r^{n}),\text{ so }s = a \frac{1-r^{n}}{1-r} \quad \text{(if } r \neq 1 \text{)}.
| |
| \end{align}
| |
| </math>
| |
| | |
| As ''n'' goes to infinity, the absolute value of ''r'' must be less than one for the series to converge. The sum then becomes
| |
| | |
| :<math>a+ar+ar^2+ar^3+ar^4+\cdots = \sum_{k=0}^\infty ar^k = \frac{a}{1-r} \Leftrightarrow |r|<1 </math>
| |
| | |
| When {{nowrap|1= ''a'' = 1}}, this simplifies to:
| |
| | |
| :<math>1 \,+\, r \,+\, r^2 \,+\, r^3 \,+\, \cdots \;=\; \frac{1}{1-r},</math>
| |
| | |
| the left-hand side being a geometric series with common ratio ''r''. We can derive this formula: | |
| | |
| :<math>
| |
| \begin{align}
| |
| &\text{Let }s = 1 + r + r^2 + r^3 + \cdots. \\[4pt]
| |
| &\text{Then }rs = r + r^2 + r^3 + \cdots. \\[4pt]
| |
| &\text{Then }s - rs = 1,\text{ so }s(1 - r) = 1,\text{ and thus }s = \frac{1}{1-r}.
| |
| \end{align}
| |
| </math>
| |
| | |
| The general formula follows if we multiply through by ''a''.
| |
| | |
| The formula holds true for complex "r", with the same restrictions (modulus of "r" is strictly less than one).
| |
| | |
| ===Proof of convergence===
| |
| We can prove that the geometric series [[convergent series|converges]] using the sum formula for a [[geometric progression]]:
| |
| :<math>\begin{align}
| |
| &1 \,+\, r \,+\, r^2 \,+\, r^3 \,+\, \cdots \\[3pt]
| |
| &=\; \lim_{n\rightarrow\infty} \left(1 \,+\, r \,+\, r^2 \,+\, \cdots \,+\, r^n\right) \\
| |
| &=\; \lim_{n\rightarrow\infty} \frac{1-r^{n+1}}{1-r}
| |
| \end{align}</math>
| |
| Since (1 + ''r'' + ''r''<sup>2</sup> + ... + ''r''<sup>''n''</sup>)(1−''r'') = 1−''r''<sup>''n''+1</sup> and {{nowrap| ''r''<sup>''n''+1</sup> → 0}} for | ''r'' | < 1.
| |
| | |
| Convergence of geometric series can also be demonstrated by rewriting the series as an equivalent [[telescoping series]]. Consider the function g(K) = (r^(K+1))/(1-r). Note that: r = g(0) - g(1), r^2 = g(1) - g(2), r^3 = g(2) - g(3), . . . Thus:
| |
| S = r + r^2 + r^3 + . . . = (g(0) - g(1)) + (g(1) - g(2)) + (g(2) - g(3)) + . . .
| |
| If |r|<1, then g(K) -> 0 as K -> infinity, and so S converges to g(0) = r/(1-r).
| |
| | |
| ==Applications==
| |
| ===Repeating decimals===
| |
| {{main|Repeating decimal}}
| |
| A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example:
| |
| | |
| :<math>0.7777\ldots \;=\; \frac{7}{10} \,+\, \frac{7}{100} \,+\, \frac{7}{1000} \,+\, \frac{7}{10000} \,+\, \cdots.</math>
| |
| | |
| The formula for the sum of a geometric series can be used to convert the decimal to a fraction:
| |
| | |
| :<math>0.7777\ldots \;=\; \frac{a}{1-r} \;=\; \frac{7/10}{1-1/10} \;=\; \frac{7}{9}.</math>
| |
| | |
| The formula works not only for a single repeating figure, but also for a repeating group of figures. For example: | |
| | |
| :<math>0.123412341234\ldots \;=\; \frac{a}{1-r} \;=\; \frac{1234/10000}{1-1/10000} \;=\; \frac{1234}{9999}.</math>
| |
| | |
| Note that every series of repeating consecutive decimals can be conveniently simplified with the following:
| |
| | |
| :<math>0.09090909\ldots \;=\; \frac{09}{99} \;=\; \frac{1}{11}.</math>
| |
| | |
| :<math>0.143814381438\ldots \;=\; \frac{1438}{9999}.</math>
| |
| | |
| :<math>0.9999\ldots \;=\; \frac{9}{9} \;=\; 1.</math>
| |
| | |
| That is, a repeating decimal with repeat length {{math|n}} is equal to the quotient of the repeating part (as an integer) and {{math|10<sup>n</sup> - 1}}.
| |
| | |
| ===Archimedes' quadrature of the parabola===
| |
| [[Image:Parabolic Segment Dissection.svg|thumb|300px|Archimedes' dissection of a parabolic segment into infinitely many triangles]]
| |
| {{main|The Quadrature of the Parabola}}
| |
| [[Archimedes]] used the sum of a geometric series to compute the area enclosed by a [[parabola]] and a straight line. His method was to dissect the area into an infinite number of triangles.
| |
| | |
| Archimedes' Theorem states that the total area under the parabola is 4/3 of the area of the blue triangle.
| |
| | |
| Archimedes determined that each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 the area of a green triangle, and so forth.
| |
| | |
| Assuming that the blue triangle has area 1, the total area is an infinite sum:
| |
| | |
| :<math>1 \,+\, 2\left(\frac{1}{8}\right) \,+\, 4\left(\frac{1}{8}\right)^2 \,+\, 8\left(\frac{1}{8}\right)^3 \,+\, \cdots.</math>
| |
| | |
| The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives | |
| | |
| :<math>1 \,+\, \frac{1}{4} \,+\, \frac{1}{16} \,+\, \frac{1}{64} \,+\, \cdots.</math>
| |
| | |
| This is a geometric series with common ratio {{nowrap|1/4}} and the fractional part is equal to {{nowrap|1/3:}}
| |
| | |
| <math>\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. \;</math>
| |
| | |
| The sum is
| |
| :<math>\frac{1}{1 -r}\;=\;\frac{1}{1 -\frac{1}{4}}\;=\;\frac{4}{3}.</math> Q.E.D.
| |
| | |
| This computation uses the [[method of exhaustion]], an early version of [[Integral|integration]]. In modern [[calculus]], the same area could be found using a [[definite integral]].
| |
| | |
| ===Fractal geometry===
| |
| [[File:Koch Snowflake Triangles.png|thumb|The interior of the [[Koch snowflake]] is a union of infinitely many triangles.]]
| |
| In the study of [[fractal]]s, geometric series often arise as the [[perimeter]], [[area]], or [[volume]] of a [[self-similarity|self-similar]] figure.
| |
| | |
| For example, the area inside the [[Koch snowflake]] can be described as the union of infinitely many [[equilateral triangle]]s (see figure). Each side of the green triangle is exactly 1/3 the size of a side of the large blue triangle, and therefore has exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking the blue triangle as a unit of area, the total area of the snowflake is
| |
| | |
| :<math>1 \,+\, 3\left(\frac{1}{9}\right) \,+\, 12\left(\frac{1}{9}\right)^2 \,+\, 48\left(\frac{1}{9}\right)^3 \,+\, \cdots.</math>
| |
| | |
| The first term of this series represents the area of the blue triangle, the second term the total area of the three green triangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this series is geometric with constant ratio ''r'' = 4/9. The first term of the geometric series is ''a'' = 3(1/9) = 1/3, so the sum is
| |
| | |
| :<math>1\,+\,\frac{a}{1-r}\;=\;1\,+\,\frac{\frac{1}{3}}{1-\frac{4}{9}}\;=\;\frac{8}{5}.</math>
| |
| | |
| Thus the Koch snowflake has 8/5 of the area of the base triangle.
| |
| | |
| ===Zeno's paradoxes===
| |
| {{main|Zeno's paradoxes}}
| |
| The convergence of a geometric series reveals that a sum involving an infinite number of summands can indeed be finite, and so allows one to resolve many of [[Zeno of Elea|Zeno]]'s paradoxes. For example, Zeno's dichotomy paradox maintains that movement is impossible, as one can divide any finite path into an infinite number of steps wherein each step is taken to be half the remaining distance. Zeno's mistake is in the assumption that the sum of an infinite number of finite steps cannot be finite. This is of course not true, as evidenced by the convergence of the geometric series with <math>r = 1/2</math>.
| |
| | |
| ===Euclid===
| |
| | |
| Book IX, Proposition 35<ref>{{cite web|url=http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX35.html |title=Euclid's Elements, Book IX, Proposition 35 |publisher=Aleph0.clarku.edu |date= |accessdate=2013-08-01}}</ref> of [[Euclid's Elements]] expresses the partial sum of a geometric series in terms of members of the series. It is equivalent to the modern formula.
| |
| | |
| ===Economics===
| |
| {{main|Time value of money}}
| |
| In [[economics]], geometric series are used to represent the [[present value]] of an [[Annuity (finance theory)|annuity]] (a sum of money to be paid in regular intervals).
| |
| | |
| For example, suppose that a payment of $100 will be made to the owner of the annuity once per year (at the end of the year) in [[perpetuity]]. Receiving $100 a year from now is worth less than an immediate $100, because one cannot [[investment|invest]] the money until one receives it. In particular, the present value of $100 one year in the future is $100 / (1 + <math>I</math> ), where <math>I</math> is the yearly interest rate.
| |
| | |
| Similarly, a payment of $100 two years in the future has a present value of $100 / (1 + <math>I</math>)<sup>2</sup> (squared because two years' worth of interest is lost by not receiving the money right now). Therefore, the present value of receiving $100 per year in perpetuity is
| |
| | |
| :<math>\sum_{n=1}^\infty \frac{\$100}{(1+I)^n},</math>
| |
| | |
| which is the infinite series:
| |
| | |
| :<math>\frac{\$ 100}{(1+I)} \,+\, \frac{\$ 100}{(1+I)^2} \,+\, \frac{\$ 100}{(1+I)^3} \,+\, \frac{\$ 100}{(1+I)^4} \,+\, \cdots.</math>
| |
| | |
| This is a geometric series with common ratio 1 / (1 + <math>I</math> ). The sum is the first term divided by (one minus the common ratio):
| |
| | |
| :<math>\frac{\$ 100/(1+I)}{1 - 1/(1+I)} \;=\; \frac{\$ 100}{I}.</math>
| |
| | |
| For example, if the yearly interest rate is 10% (<math>I</math> = 0.10), then the entire annuity has a present value of $100 / 0.10 = $1000.
| |
| | |
| This sort of calculation is used to compute the [[annual percentage rate|APR]] of a loan (such as a [[mortgage loan]]). It can also be used to estimate the present value of expected [[Dividend|stock dividends]], or the [[terminal value]] of a [[security (finance)|security]].
| |
| | |
| ===Geometric power series===
| |
| | |
| The formula for a geometric series
| |
| | |
| :<math>\frac{1}{1-x}=1+x+x^2+x^3+x^4+\cdots</math>
| |
| | |
| can be interpreted as a [[power series]] in the [[Taylor's theorem]] sense, converging where <math>|x|<1</math>. From this, one can extrapolate to obtain other power series. For example,
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \tan^{-1}(x)&=\int\frac{dx}{1+x^2}=\int\frac{dx}{1-(-x^2)}=\int\left(1 + \left(-x^2\right) + \left(-x^2\right)^2 + \left(-x^2\right)^3 + \cdots\right)dx\\
| |
| &=\int\left(1-x^2+x^4-x^6+\cdots\right)dx=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots=\sum^{\infin}_{n=0} \frac{(-1)^n}{2n+1} x^{2n+1}
| |
| \end{align}</math>
| |
| | |
| By differentiating the geometric series, one obtains the variant
| |
| | |
| :<math>\frac{1}{(1-x)^2} = \sum^{\infin}_{n=1}n x^{n-1}\quad\text{ for }|x| < 1.</math> | |
| | |
| ==See also==
| |
| *[[Asymptote]]
| |
| *[[Divergent geometric series]]
| |
| *[[Geometric progression]]
| |
| *[[Neumann series]]
| |
| *[[Generalized hypergeometric function]]
| |
| *[[Ratio test]]
| |
| *[[Root test]]
| |
| *[[Series (mathematics)]]
| |
| *[[Tower of Hanoi]]
| |
| *[[0.999...]]
| |
| | |
| ===Specific geometric series===
| |
| *[[Grandi's series]]: 1 − 1 + 1 − 1 + · · ·
| |
| *[[1 + 2 + 4 + 8 + · · ·]]
| |
| *[[1 − 2 + 4 − 8 + · · ·]]
| |
| *[[1/2 + 1/4 + 1/8 + 1/16 + · · ·]]
| |
| *[[1/2 − 1/4 + 1/8 − 1/16 + · · ·]]
| |
| *[[1/4 + 1/16 + 1/64 + 1/256 + · · ·]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| {{refbegin}}
| |
| * James Stewart (2002). ''Calculus'', 5th ed., Brooks Cole. ISBN 978-0-534-39339-7
| |
| * Larson, Hostetler, and Edwards (2005). ''Calculus with Analytic Geometry'', 8th ed., Houghton Mifflin Company. ISBN 978-0-618-50298-1
| |
| * Roger B. Nelsen (1997). ''Proofs without Words: Exercises in Visual Thinking'', The Mathematical Association of America. ISBN 978-0-88385-700-7
| |
| * {{cite journal |author=Andrews, George E. |title=The geometric series in calculus|journal=The American Mathematical Monthly |volume=105 |issue=1 | year=1998 |pages=36–40 |doi=10.2307/2589524 |jstor=2589524 |publisher=Mathematical Association of America}}
| |
| | |
| ===History and philosophy===
| |
| * C. H. Edwards, Jr. (1994). ''The Historical Development of the Calculus'', 3rd ed., Springer. ISBN 978-0-387-94313-8.
| |
| * {{cite journal |doi=10.2307/2691014 |author=Swain, Gordon and Thomas Dence |title=Archimedes' Quadrature of the Parabola Revisited |journal=Mathematics Magazine |volume=71 |issue=2 |date=April 1998 |pages=123–30 |jstor=2691014}}
| |
| * [[Eli Maor]] (1991). ''To Infinity and Beyond: A Cultural History of the Infinite'', Princeton University Press. ISBN 978-0-691-02511-7
| |
| * Morr Lazerowitz (2000). ''The Structure of Metaphysics (International Library of Philosophy)'', Routledge. ISBN 978-0-415-22526-7
| |
| | |
| ===Economics===
| |
| * Carl P. Simon and Lawrence Blume (1994). ''Mathematics for Economists'', W. W. Norton & Company. ISBN 978-0-393-95733-4
| |
| * Mike Rosser (2003). ''Basic Mathematics for Economists'', 2nd ed., Routledge. ISBN 978-0-415-26784-7
| |
| | |
| ===Biology===
| |
| * Edward Batschelet (1992). ''Introduction to Mathematics for Life Scientists'', 3rd ed., Springer. ISBN 978-0-387-09648-3
| |
| * Richard F. Burton (1998). ''Biology by Numbers: An Encouragement to Quantitative Thinking'', Cambridge University Press. ISBN 978-0-521-57698-7
| |
| | |
| ===Computer science===
| |
| * John Rast Hubbard (2000). ''Schaum's Outline of Theory and Problems of Data Structures With Java'', McGraw-Hill. ISBN 978-0-07-137870-3
| |
| {{refend}}
| |
| | |
| ==External links==
| |
| * {{springer|title=Geometric progression|id=p/g044290}}
| |
| * {{MathWorld|title=Geometric Series|urlname=GeometricSeries}}
| |
| * {{PlanetMath|title=Geometric Series|urlname=InfiniteGeometricSeries}}
| |
| * {{cite web|last=Peppard|first=Kim|title=College Algebra Tutorial on Geometric Sequences and Series|url=http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut54d_geom.htm|publisher=West Texas A&M University}}
| |
| * {{cite web|last=Casselman|first=Bill|title=A Geometric Interpretation of the Geometric Series|format=Applet|url=http://merganser.math.gvsu.edu/calculus/summation/geometric.html}}
| |
| * [http://demonstrations.wolfram.com/GeometricSeries/ "Geometric Series"] by Michael Schreiber, [[Wolfram Demonstrations Project]], 2007.
| |
| | |
| {{DEFAULTSORT:Geometric Series}}
| |
| [[Category:Geometric series]]
| |
| [[Category:Calculus]]
| |
| [[Category:Articles containing proofs]]
| |
| | |
| [[he:סדרה הנדסית]]
| |
| [[zh:等比数列]]
| |