Geometric series
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 ...
is a geometric series with common ratio 2. This is the same as 2 * 2
^{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

which is valid for all
natural numbers 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
An
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

which is valid whenever 
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 padic numbers if x_{p} < 1.
Also useful to mention:

which can be seen as
x times the derivative of the infinite geometric series. This formula only works for 
x < 1, as well.
See also: infinite series