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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
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.
A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3.
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 .
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.
The nth element of an arithmetico-geometric sequence is the product of the nth element of an arithmetic sequence and the nth element of a geometric sequence. [1] An arithmetico-geometric series is a sum of terms that are the elements of an arithmetico-geometric sequence. Arithmetico-geometric sequences and series arise in various applications ...
A famous example of an application of this test is the alternating harmonic series = + = + +, which is convergent per the alternating series test (and its sum is equal to ), though the series formed by taking the absolute value of each term is the ordinary harmonic series, which is divergent.
A similar phenomenon occurs with the divergent geometric series + + (Grandi's series), where a series of integers appears to have the non-integer sum . These examples illustrate the potential danger in applying similar arguments to the series implied by such recurring decimals as 0.111 … {\displaystyle 0.111\ldots } and most notably 0.999 ...