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This corresponds to a similar property of sums of terms of a finite arithmetic sequence: the sum of an arithmetic sequence is the number of terms times the arithmetic mean of the first and last individual terms. This correspondence follows the usual pattern that any arithmetic sequence is a sequence of logarithms of terms of a geometric ...
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 .
First six summands drawn as portions of a square. 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.
where i is the index of summation; a i is an indexed variable representing each term of the sum; m is the lower bound of summation, and n is the upper bound of summation. The "i = m" under the summation symbol means that the index i starts out equal to m. The index, i, is incremented by one for each successive term, stopping when i = n. [b ...
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 .
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
For instance, rearranging the terms of the alternating harmonic series so that each positive term of the original series is followed by two negative terms of the original series rather than just one yields [34] + + + = + + + = + + + = (+ + +), which is times the original series, so it would have a sum of half of the natural logarithm of 2. By ...
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (,,, …) defines a series S that is denoted = + + + = =. The n th partial sum S n is the sum of the first n terms of the sequence; that is,