Encyclopedia > Geometric series

  Article Content

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 * 2x where x is increacing 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

<math>\sum_{k=m}^n x^k=\frac{x^{n+1}-x^m}{x-1}</math>
which is valid for all natural numbers mn 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 to mention:

<math>\sum_{k=0}^\infty k\cdot x^k=\frac{x}{(1-x)^2}</math>
This formula only works for |x| < 1, as well.

See also: infinite series



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
My Sister Eileen

... George Tobias[?], June Havoc, Larry Fine[?], Curly Howard[?], Moe Howard[?], (the Three Stooges), Kirk Alyn[?], Arnold Stang[?] and Forrest Tucker[?]. It was ...