A

**geometric series**is a sum of terms in which two successive terms always have the same ratio. For example,

- 4 + 8 + 16 + 32 + 64 + 128 + 256 ...

^{x}where x is increasing by one for each number. It is called a geometric series because it occurs when comparing the length, area, volume, etc. of a shape in different dimensions.

The sum of a geometric series can be computed quickly with the formula

*m*≤

*n*and all numbers

*x*≠ 1 (or more generally, for all elements

*x*in a ring such that

*x*- 1 is invertible). This formula can be verified by multiplying both sides with

*x*- 1 and simplifying.

Using the formula, we
can determine the above sum: (2^{9} - 2^{2})/(2 - 1) = 508. The formula is also extremely useful in calculating annuities: suppose you put $2,000 in the bank every year, and the money earns interest at an annual rate of 5%. How much money do you have after 6 years?

- 2,000 · 1.05
^{6}+ 2,000 · 1.05^{5}+ 2,000 · 1.05^{4}+ 2,000 · 1.05^{3}+ 2,000 · 1.05^{2}+ 2,000 · 1.05^{1} - = 2,000 · (1.05
^{7}- 1.05)/(1.05 - 1) - = 14,284.02

**infinite geometric series**is an infinite series whose successive terms have a common ratio. Such a series converges if and only if the absolute value of the common ratio is less than one; its value can then be computed with the formula

*x*| < 1; it is a consequence of the above formula for finite geometric series by taking the limit for

*n*→∞.

This last formula is actually valid in every Banach algebra, as long as the norm of *x* is less than one, and also in the field of *p*-adic numbers if |*x*|_{p} < 1.

Also useful to mention:

*x*times the derivative of the infinite geometric series. This formula only works for |

*x*| < 1, as well.

**See also**: infinite series