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
where
a is the first term, and
r is the constant ratio. If the first term
is
2 and the constant ratio is
,
the geometric series is
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
.
The partial sum of the first four terms of the series
is
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
.
The sum for the geometric series
is
.
Properties
 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. 
References
 geometric series. http://dictionary.reference.com/browse/geometric+series. Dictionary.com Unabridged. Random House, Inc.. (Accessed: 20100206). http://dictionary.reference.com/browse/geometric+series.
 Bettinger, Alvin K. and Englund, John A.. Algebra And Trigonometry, pp 265267. International Textbook Company, January 1963. (Accessed: 20100206). 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 174177. D. Appleton and Company, 1910. (Accessed: 20100206). http://www.archive.org/stream/secondcourseinel00younrich#page/174/mode/1up/search/progression.
More Information
 Knaust, Helmut. The Geometric Series. sosmath.com. SOS Math. 20091201. http://www.sosmath.com/calculus/geoser/geoser01.html.
 Stapel, Elizabeth. Geometric Series. purplemath.com. PurpleMath. 20091201. http://www.purplemath.com/modules/series5.htm.
Printed Resources
Cite this article as:
Geometric Series. 20100206. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/g/geometricseries.html.
Translations
Image Credits
Revision History
20100206: Added "References" (
McAdams, David.)
20091201: Initial version. (
McAdams, David.)