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The geometric series on the real line. In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
Using sigma summation notation the sum of the first m terms of the series can be expressed as = (). The infinite series diverges, meaning that its sequence of partial sums, (1, −1, 2, −2, 3, ...), does not tend towards any finite limit.
The same geometric strategy also works for triangles, as in the figure on the right: [4] if the large triangle has area 1, then the largest black triangle has area 1 / 4 , and so on. The figure as a whole has a self-similarity between the large triangle and its upper sub-triangle.
Greek mathematician Archimedes produced the first known summation of an infinite series with a method that is still used in the area of calculus today. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, [5] and gave a remarkably accurate approximation of π. [80] [81]
3. Often used for denoting other types of similarity, for example, matrix similarity or similarity of geometric shapes. 4. Standard notation for an equivalence relation. 5. In probability and statistics, may specify the probability distribution of a random variable.
The geometric series is an infinite series derived from a special type of sequence called a geometric progression.This means that it is the sum of infinitely many terms of geometric progression: starting from the initial term , and the next one being the initial term multiplied by a constant number known as the common ratio .
For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of − + 1 / 12 , which is expressed by a famous formula: [ 2 ]