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In mathematics, a geometric series is a series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, the series 1 2 + 1 4 + 1 8 + ⋯ {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{8}}+\cdots } is a geometric series with common ratio 1 2 {\displaystyle {\tfrac {1 ...
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 .
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.
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.
In mathematics, the nth-term test for divergence [1] is a simple test for the divergence of an infinite series: If or if the limit does not exist, then = diverges. Many authors do not name this test or give it a shorter name. [2]
6 ; 3 + 3 ; 1 + 4 + 1 ; 1 + 1 + 1 + 1 + 1 + 1 The number of ways of writing n as an ordered sum in which each term is odd and greater than 1 is equal to P ( n − 5). For example, P (6) = 4, and there are 4 ways to write 11 as an ordered sum in which each term is odd and greater than 1:
If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge. Root test or nth root test. Suppose that the terms of the sequence in question are non-negative. Define r as follows:
where is the number of terms in the progression and is the common difference between terms. The formula is essentially the same as the formula for the standard deviation of a discrete uniform distribution , interpreting the arithmetic progression as a set of equally probable outcomes.