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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 .
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. If 1 < p, then the nth-term test is inconclusive, but the series is convergent by the integral test for convergence.
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 ...
Suppose that (f n) is a sequence of real- or complex-valued functions defined on a set A, and that there is a sequence of non-negative numbers (M n) satisfying the conditions | f n ( x ) | ≤ M n {\displaystyle |f_{n}(x)|\leq M_{n}} for all n ≥ 1 {\displaystyle n\geq 1} and all x ∈ A {\displaystyle x\in A} , and
These homotopy classes form a group, called the n-th homotopy group, (), of the given space X with base point. Topological spaces with differing homotopy groups are never homeomorphic , but topological spaces that are not homeomorphic can have the same homotopy groups.
More formally, one describes it in terms of functions on closed sets of points. If we let d A denote the dilation by a factor of 1 / 2 about a point A, then the Sierpiński triangle with corners A, B, and C is the fixed set of the transformation d A ∪ d B ∪ d C {\displaystyle d_{\mathrm {A} }\cup d_{\mathrm {B} }\cup d_{\mathrm ...
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation.
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