# 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 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

 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

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

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.

### 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.)