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 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 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
- <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
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