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  2. Alternating series - Wikipedia

    en.wikipedia.org/wiki/Alternating_series

    The geometric series ⁠ 1 / 2 ⁠ − ⁠ 1 / 4 ⁠ + ⁠ 1 / 8 ⁠ − ⁠ 1 / 16 ⁠ + ⋯ sums to ⁠ 1 / 3 ⁠.. The alternating harmonic series has a finite sum but the harmonic series does not.

  3. 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 .

  4. 1 − 2 + 4 − 8 + ⋯ - ⋯ - Wikipedia

    en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_4_%E2%88...

    In mathematics, 1 − 2 + 4 − 8 + ⋯ is the infinite series whose terms are the successive powers of two with alternating signs. 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. So in the usual sense it has no sum.

  5. Geometric progression - Wikipedia

    en.wikipedia.org/wiki/Geometric_progression

    When the common ratio of a geometric sequence is negative, the sequence's terms alternate between positive and negative; this is called an alternating sequence. For instance the sequence 1, −3, 9, −27, 81, −243, ... is an alternating geometric sequence with an initial value of 1 and a common ratio of −3.

  6. 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.

  7. Leibniz formula for π - Wikipedia

    en.wikipedia.org/wiki/Leibniz_formula_for_π

    In mathematics, the Leibniz formula for π, named after Gottfried Wilhelm Leibniz, states that = + + = = +,. an alternating series.. It is sometimes called the Madhava–Leibniz series as it was first discovered by the Indian mathematician Madhava of Sangamagrama or his followers in the 14th–15th century (see Madhava series), [1] and was later independently rediscovered by James Gregory in ...

  8. 1/2 − 1/4 + 1/8 − 1/16 + ⋯ - ⋯ - Wikipedia

    en.wikipedia.org/wiki/1/2_%E2%88%92_1/4_%2B_1/8...

    In mathematics, the infinite series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely. It is a geometric series whose first term is ⁠ 1 / 2 ⁠ and whose common ratio is − ⁠ 1 / 2 ⁠, so its sum is

  9. Series (mathematics) - Wikipedia

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

    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 ...