Pronunciation: /ˌdʒi əˈmɛt rɪk ˈsɪər iz/ Explain
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
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.|
- geometric series. Dictionary.com Unabridged. Random House, Inc.. Last Accessed 8/6/2018. http://dictionary.reference.com/browse/geometric+series.
- Bettinger, Alvin K. and Englund, John A.. Algebra And Trigonometry. pp 265-267. www.archive.org. International Textbook Company. January 1963. Last Accessed 8/6/2018. http://www.archive.org/stream/algebraandtrigon033520mbp#page/n282/mode/1up/search/progression. Buy the book
- 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 8/6/2018. http://www.archive.org/stream/secondcourseinel00younrich#page/174/mode/1up/search/progression. Buy the book
- 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. 7/11/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/g/geometricseries.html.
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.)