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Geometric Series

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A geometric series sum_(k)a_k is a series for which the ratio of each two consecutive terms a_(k+1)/a_k is a constant function of the summation index k. The more general case of the ratio a rational function of the summation index k produces a series called a hypergeometric series.

For the simplest case of the ratio a_(k+1)/a_k=r equal to a constant r, the terms a_k are of the form a_k=a_0r^k. Letting a_0=1, the geometric sequence {a_k}_(k=0)^n with constant |r|<1 is given by

S_n=sum_(k=0)^na_k=sum_(k=0)^nr^k
(1)

is given by

S_n=sum_(k=0)^nr^k=1+r+r^2+...+r^n.
(2)

Multiplying both sides by r gives

rS_n=r+r^2+r^3+...+r^(n+1),
(3)

and subtracting (3) from (2) then gives

(1-r)S_n=(1+r+r^2+...+r^n)-(r+r^2+r^3+...+r^(n+1))
(4)
=1-r^(n+1),
(5)

so

S_n=sum_(k=0)^nr^k=(1-r^(n+1))/(1-r).
(6)

For -1<r<1, the sum converges as n->infty,in which case

S=S_infty=sum_(k=0)^inftyr^k=1/(1-r)
(7)

Similarly, if the sums are taken starting at k=1 instead of k=0,

sum_(k=1)^(n)r^k=(r(1-r^n))/(1-r)
(8)
sum_(k=1)^(infty)r^k=r/(1-r),
(9)

the latter of which is valid for |r|<1.

See also

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.

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Geometric Series

Cite this as:

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

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