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