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Examples of a geometric sequence are powers r k of a fixed non-zero number r, such as 2 k and 3 k. The general form of a geometric sequence is , , , , , … where r is the common ratio and a is the initial value. The sum of a geometric progression's terms is called a geometric series.
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 .
Partial summation of a sequence is an example of a ... In general, a geometric series ... who also gave a general form for the remainder of the Maclaurin formula. The ...
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
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, such as the computation of expected values in probability theory , especially in Bernoulli processes .
The next table provides examples of closed-form formulas for the component sequences found computationally (and subsequently proved correct in the cited references [22]) in several special cases of the prescribed sequences, j n, generated by the general expansions of the J-fractions defined in the first subsection.
The triangular number sequence is the representation of the numbers in the form of equilateral triangle arranged in a series or sequence. These numbers are in a sequence of 1, 3, 6, 10, 15, 21, 28, 36, 45, and so on.
The geometric distribution is the discrete probability distribution that describes when the first success in an infinite sequence of independent and identically distributed Bernoulli trials occurs. Its probability mass function depends on its parameterization and support .