Geometric Series

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 $k$, say $r$. Then the terms $a_k$ are of the form $a_k=a_0 r^k$, so $a_{k+1}/a_k=r$. If $\{a_k\}$, with $k=1$, 2, ..., is a Geometric Sequence with multiplier $-1<r<1$ and $a_0=1$, then the geometric series

$$S_n=\sum_{k=0}^n a_k = \sum_{k=0}^n r^k$$ (1)

is given by

$$S_n\equiv \sum_{k=0}^n r^k = 1+r+r^2+\ldots+r^n,$$ (2)

so

$$rS_n = r+r^2+r^3+\ldots+r^{n+1}.$$ (3)

Subtracting

$$(1-r)S_n = (1+r+r^2+\ldots+r^n)-(r+r^2+r^3+\ldots+r^{n+1})$$

$$= 1-r^{n+1},$$ (4)

so

$$S_n\equiv\sum_{k=0}^n r^k = {1-r^{n+1}\over 1-r}.$$ (5)

As $n\to \infty$, then

$$S \equiv S_\infty = \sum_{k=0}^\infty r^k = {1\over 1-r}$$ (6)

for $-1<r<1$.

See also Arithmetic Series, Gabriel's Staircase, Harmonic Series, Hypergeometric Series, Wheat and Chessboard Problem

References

Abramowitz, M. and Stegun, C. 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.