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# Geometric Series

A geometric series is a series for which the ratio of each two consecutive terms is a constant function of the summation index . The more general case of the ratio a rational function of the summation index produces a series called a hypergeometric series.

For the simplest case of the ratio equal to a constant , the terms are of the form . Letting , the geometric sequence with constant is given by

 (1)

is given by

 (2)

Multiplying both sides by gives

 (3)

and subtracting (3) from (2) then gives

 (4) (5)

so

 (6)

For , the sum converges as ,in which case

 (7)

Similarly, if the sums are taken starting at instead of ,

 (8) (9)

the latter of which is valid for .

Arithmetic Series, Gabriel's Staircase, Harmonic Series, Hypergeometric Series, St. Ives Problem, Wheat and Chessboard Problem Explore this topic in the MathWorld classroom

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## References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 278-279, 1985.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 8, 1987.Courant, R. and Robbins, H. "The Geometric Progression." §1.2.3 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 13-14, 1996.Pappas, T. "Perimeter, Area & the Infinite Series." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 134-135, 1989.

Geometric Series

## Cite this as:

Weisstein, Eric W. "Geometric Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GeometricSeries.html