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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 .
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 ...
The nth partial sum of the series is the ... f decay quickly enough for the remainder terms in the Euler–Maclaurin formula to ... means for geometric series, ...
All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a homogeneous linear difference equation with constant coefficients. Some specific examples that are close, in some sense, to the Fibonacci sequence include:
In mathematics, 1 + 2 + 4 + 8 + ⋯ is the infinite series whose terms are the successive powers of two. As a geometric series , it is characterized by its first term, 1, and its common ratio , 2. As a series of real numbers it diverges to infinity , so the sum of this series is infinity.
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 .
The more general class of p-series, =, exemplifies the possible results of the test: If p ≤ 0, then the nth-term test identifies the series as divergent. If 0 < p ≤ 1, then the nth-term test is inconclusive, but the series is divergent by the integral test for convergence.