Pronunciation: /ˌdʒi əˈmɛt rɪk ˈsɪər iz/ ?
A geometric series is a series with a
The common ratio is a
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
is the first term, and r
is the constant ratio. If the first term
and the constant ratio is
the geometric series is
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
The partial sum of the first four terms of the series
The sum of a geometric series is the sum of an
If the common ratio is less than 1, the sum can be calculated using the formula
The sum for the geometric series
- If 0 < r < 1, successive terms become smaller and smaller, getting closer to zero. The sum can be found using the formula
- 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
|Manipulative 1: Geometric series. Created with GeoGebra.|
- 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.
- 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.
- 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.
- 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.
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.
2010-02-06: Added "References" (McAdams, David.
2009-12-01: Initial version. (McAdams, David.