Geometric Series

Pronunciation: /ˌdʒi əˈmɛt rɪk ˈsɪər iz/ ?

A geometric series is a series with a common ratio between successive terms. The common ratio is a constant. For example, the series 1 + 2 + 4 + 8 + ... is a geometric series. The constant ratio between each term is 2: (1·2 = 2, 2·2 = 4; 4·2 = 8, ...).

A geometric series S can written in the notation S = a + ar + ar^2 + ar^3 + ... where a is the first term, and r is the constant ratio. If the first term is 2 and the constant ratio is 1/3, the geometric series is
a = 2; r = 1/3; S = a+ar+ar^2+ar^3+...; S = 2+2(1/3)+2(1/3)^2+2(1/3)^3+...; S = 2+(2/3)+(2/9)+(2/27)+...

Sum

The sum of a geometric series is the sum of each of the terms of the geometric series. A partial sum is a sum of the first n terms of the geometric series. The notation for a partial sum of the first n terms of a geometric series is written s sub n. The partial sum of the first four terms of the series 1+(1/2)+(1/4)+(1/8)+... is
S=1+(1/2)+(1/4)+(1/8)+...=(8/8)+(4/8)+(2/8)+(1/8)=(8+4+2+1)/8=15/8

The sum of a geometric series is the sum of an infinite number of addends. If the common ratio is less than 1, the sum can be calculated using the formula
s=a*1/(1-r).
The sum for the geometric series 1+(1/2)+(1/4)+(1/8)+... is
s=a*1/(1-r)=1*(1)/(1-(1/2))=1/(1/2)=2.

Properties

  • If 0 < r < 1, successive terms become smaller and smaller, getting closer to zero. The sum can be found using the formula s=a*1/(1-r)
  • If r = 1, all terms are the same value. The sum does not exist.
  • If r > 1, successive terms become bigger and bigger, approaching infinity. The sum does not exist.
  • if r = -1, successive terms are additive inverses (a, -a, a, -a, ...), the partial sums of the terms oscilates. The sum does not exist.

Graphical Representation of Geometric Series

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
Manipulative 1: Geometric series. Created with GeoGebra.

References

  1. geometric series. http://dictionary.reference.com/browse/geometric+series. Dictionary.com Unabridged. Random House, Inc.. (Accessed: 2010-02-06). http://dictionary.reference.com/browse/geometric+series.
  2. Bettinger, Alvin K. and Englund, John A.. Algebra And Trigonometry, pp 265-267. International Textbook Company, January 1963. (Accessed: 2010-02-06). http://www.archive.org/stream/algebraandtrigon033520mbp#page/n282/mode/1up/search/progression.
  3. Young, J. W. A. and Jackson, Lambert L.. A Second Course in Elementary Algebra, pp 174-177. D. Appleton and Company, 1910. (Accessed: 2010-02-06). http://www.archive.org/stream/secondcourseinel00younrich#page/174/mode/1up/search/progression.

More Information

  • Knaust, Helmut. The Geometric Series. sosmath.com. SOS Math. 2009-12-01. http://www.sosmath.com/calculus/geoser/geoser01.html.
  • Stapel, Elizabeth. Geometric Series. purplemath.com. PurpleMath. 2009-12-01. http://www.purplemath.com/modules/series5.htm.

Printed Resources

Cite this article as:


Geometric Series. 2010-02-06. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/g/geometricseries.html.

Translations

Image Credits

Revision History


2010-02-06: Added "References" (McAdams, David.)
2009-12-01: Initial version. (McAdams, David.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2005-2011 Life is a Story Problem LLC. All rights reserved.
Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License