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Geometric series

mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence.

Thus without loss of generality a geometric sequence can be written as <math>a(r^0,r^1,r^2,r^3,...)\,<math> where r is the common ratio and a is a scale factor. Thus the common ratio gives a family of geometric sequences whose starting value is determined by the scale factor.

For example, a sequence with a common ratio of 2 and a scale factor of 1 is

1, 2, 4, 8, 16, 32, 64, 128, 256, ....

The sum of the numbers in a geometric progression is called a geometric series. Thus the geometric series for the n terms of a geometric progression is

<math>a(r^0+r^1+r^2+r^3+...+r^{n-1})\,<math>

Multiplying <math>(1+r+r^2+r^3+...+r^{n-1})\,<math> by <math>(1-r)\,<math> equals <math>(1-r^n)\,<math> since all the other terms cancel in pairs.

Rearranging gives the general formula for the sum of a geometric series:

<math>a\sum_{k=0}^{n-1} r^k=a\frac{r^{n}-1}{r-1}<math>

An interesting relationship for a geometric series is given by:

<math>a(r^0+r^1+r^2+r^3+r^4+r^5+r^6+r^7+r^8+...) = a(r^0+r^1+r^2)(r^0+r^3+r^6+...)\,<math>

For example, (1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 +...) can be written as (1 + 2 + 4)(1 + 8 + 64 +...)

A geometric series is a sum of terms in which two successive terms always have the same ratio, i.e., the sequence of terms is a geometric sequence. 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 whose first term is a power of the common ratio can be computed quickly with the formula

<math>\sum_{k=m}^n x^k=\frac{x^{n+1}-x^m}{x-1}<math>

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: (2⁹ − 2²)/(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 · 1.05⁶ + 2,000 · 1.05⁵ + 2,000 · 1.05⁴ + 2,000 · 1.05³ + 2,000 · 1.05² + 2,000 · 1.05¹

= 2,000 · (1.05⁷ − 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

<math>\sum_{k=0}^\infty x^k=\frac{1}{1-x}<math>

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 p-adic numbers if |x|_p < 1.

Also useful is the formula

<math>\sum_{k=0}^\infty k\cdot x^k=\frac{x}{(1-x)^2}<math>

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