Geometric Series

Pronunciation: /ˌdʒi.əˈmɛt.rɪk ˈsɪər.iz/ Explain

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

Click on the points on the sliders in the upper left corner and drag them to change the figure.

Manipulative 1 - Geometric Series Created with GeoGebra.

References

  1. McAdams, David E.. All Math Words Dictionary, geometric series. 2nd Classroom edition 20150108-4799968. pg 85. Life is a Story Problem LLC. January 8, 2015. Buy the book
  2. geometric series. Merriam-Webster Online Dictionary. Merriam-Webster. Last Accessed 12/26/2018. https://www.merriam-webster.com/dictionary/geometric%20series. Buy the book
  3. Bettinger, Alvin K. and Englund, John A.. Algebra And Trigonometry. pp 265-267. www.archive.org. International Textbook Company. January 1963. Last Accessed 7/11/2018. http://www.archive.org/stream/algebraandtrigon033520mbp#page/n282/mode/1up/search/progression. Buy the book
  4. Young, J. W. A. and Jackson, Lambert L.. A Second Course in Elementary Algebra. pp 174-177. www.archive.org. D. Appleton and Company. 1910. Last Accessed 7/11/2018. http://www.archive.org/stream/secondcourseinel00younrich#page/174/mode/1up/search/progression. Buy the book

More Information

  • Knaust, Helmut. The Geometric Series. sosmath.com. SOS Math. 12/1/2009. http://www.sosmath.com/calculus/geoser/geoser01.html.

Cite this article as:

 McAdams, David E. Geometric Series. 4/21/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/g/geometricseries.html.

Image Credits

Revision History

4/21/2019: Updated equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
7/10/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
2/6/2010: Added "References". (McAdams, David E.)
12/1/2009: Initial version. (McAdams, David E.)

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