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  2. Divergent series - Wikipedia

    en.wikipedia.org/wiki/Divergent_series

    Euler summation is essentially an explicit form of analytic continuation. If a power series converges for small complex z and can be analytically continued to the open disk with diameter from ⁠ −1 / q + 1 ⁠ to 1 and is continuous at 1, then its value at q is called the Euler or (E,q) sum of the series Σa n. Euler used it before analytic ...

  3. Euler summation - Wikipedia

    en.wikipedia.org/wiki/Euler_summation

    Given a series Σa n, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original series. As well as being used to define values for divergent series, Euler summation can be used to speed the convergence of series. Euler summation can be generalized into a family of methods denoted (E, q), where q ≥ 0. The ...

  4. 1 + 2 + 3 + 4 - 3 + 4 + ... - Wikipedia

    en.wikipedia.org/wiki/Sum_of_natural_numbers

    Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable and linear cannot sum 1 + 2 + 3 + ⋯ to a finite value (see § Heuristics below). More advanced methods are required, such as zeta function regularization or Ramanujan summation.

  5. Divergence of the sum of the reciprocals of the primes

    en.wikipedia.org/wiki/Divergence_of_the_sum_of...

    While the partial sums of the reciprocals of the primes eventually exceed any integer value, they never equal an integer. One proof [6] is by induction: The first partial sum is ⁠ 1 / 2 ⁠, which has the form ⁠ odd / even ⁠. If the n th partial sum (for n ≥ 1) has the form ⁠ odd / even ⁠, then the (n + 1) st sum is

  6. Harmonic series (mathematics) - Wikipedia

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

    Because it is a divergent series, it should be interpreted as a formal sum, an abstract mathematical expression combining the unit fractions, rather than as something that can be evaluated to a numeric value. There are many different proofs of the divergence of the harmonic series, surveyed in a 2006 paper by S. J. Kifowit and T. A. Stamps. [13]

  7. 1 − 2 + 3 − 4 + ⋯ - ⋯ - Wikipedia

    en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88...

    Euler had already become famous for finding the values of these functions at positive even integers (including the Basel problem), and he was attempting to find the values at the positive odd integers (including Apéry's constant) as well, a problem that remains elusive today.

  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 / and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, although he ...

  9. Riemann zeta function - Wikipedia

    en.wikipedia.org/wiki/Riemann_zeta_function

    The Riemann zeta function ζ(z) plotted with domain coloring. [1] The pole at = and two zeros on the critical line.. The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (), is a mathematical function of a complex variable defined as () = = = + + + for ⁡ >, and its analytic continuation elsewhere.