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  2. Geometric series - Wikipedia

    en.wikipedia.org/wiki/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 .

  3. Geometric progression - Wikipedia

    en.wikipedia.org/wiki/Geometric_progression

    For example, the series + + + is a geometric series with common ratio ⁠ ⁠, which converges to the sum of ⁠ ⁠. Each term in a geometric series is the geometric mean of the term before it and the term after it, in the same way that each term of an arithmetic series is the arithmetic mean of its neighbors.

  4. Series (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Series_(mathematics)

    A summation method is any method for assigning sums to divergent series in a way that systematically extends the classical notion of the sum of a series. Summation methods include Cesàro summation, generalized Cesàro ⁠ (,) ⁠ summation, Abel summation, and Borel summation, in order of applicability to increasingly divergent series.

  5. List of mathematical series - Wikipedia

    en.wikipedia.org/wiki/List_of_mathematical_series

    An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.

  6. Arithmetico-geometric sequence - Wikipedia

    en.wikipedia.org/wiki/Arithmetico-geometric_sequence

    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 .

  7. Borel summation - Wikipedia

    en.wikipedia.org/wiki/Borel_summation

    More generally one can define summation methods slightly stronger than Borel's by taking the numbers b n to be slightly larger, for example b n = cnlog n or b n =cnlog n log log n. In practice this generalization is of little use, as there are almost no natural examples of series summable by this method that cannot also be summed by Borel's method.

  8. Basel problem - Wikipedia

    en.wikipedia.org/wiki/Basel_problem

    The sum of the series is approximately equal to 1.644934. [3] The Basel problem asks for the exact sum of this series (in closed form ), as well as a proof that this sum is correct. Euler found the exact sum to be π 2 / 6 {\displaystyle \pi ^{2}/6} and announced this discovery in 1735.

  9. Glossary of calculus - Wikipedia

    en.wikipedia.org/wiki/Glossary_of_calculus

    Geometric; Definitions; ... alternating series test Is the method used to prove that an alternating series with terms that decrease ... is called the sum of the series.