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  2. Absolute convergence - Wikipedia

    en.wikipedia.org/wiki/Absolute_convergence

    The converse is also true: if absolute convergence implies convergence in a normed space, then the space is a Banach space. If a series is convergent but not absolutely convergent, it is called conditionally convergent. An example of a conditionally convergent series is the alternating harmonic series.

  3. Modes of convergence - Wikipedia

    en.wikipedia.org/wiki/Modes_of_convergence

    In a normed vector space, one can define absolute convergence as convergence of the series (| |). Absolute convergence implies Cauchy convergence of the sequence of partial sums (by the triangle inequality), which in turn implies absolute convergence of some grouping (not reordering). The sequence of partial sums obtained by grouping is a ...

  4. Uniform absolute-convergence - Wikipedia

    en.wikipedia.org/wiki/Uniform_absolute-convergence

    A series can be uniformly convergent and absolutely convergent without being uniformly absolutely-convergent. For example, if ƒ n (x) = x n /n on the open interval (−1,0), then the series Σf n (x) converges uniformly by comparison of the partial sums to those of Σ(−1) n /n, and the series Σ|f n (x)| converges absolutely at each point by the geometric series test, but Σ|f n (x)| does ...

  5. Convergent series - Wikipedia

    en.wikipedia.org/wiki/Convergent_series

    A series is convergent (or converges) if and only if the sequence (,,, … ) {\displaystyle (S_{1},S_{2},S_{3},\dots )} of its partial sums tends to a limit ; that means that, when adding one a k {\displaystyle a_{k}} after the other in the order given by the indices , one gets partial sums that become closer and closer to a given number.

  6. Convergence tests - Wikipedia

    en.wikipedia.org/wiki/Convergence_tests

    In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series =. List of tests

  7. Convergence of Fourier series - Wikipedia

    en.wikipedia.org/wiki/Convergence_of_Fourier_series

    Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur. Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, L p spaces, summability methods and the Cesàro mean.

  8. Modes of convergence (annotated index) - Wikipedia

    en.wikipedia.org/wiki/Modes_of_convergence...

    - Absolute-convergence and convergence together unconditional convergence. - Unconditional convergence ⇏ {\displaystyle \not \Rightarrow } absolute-convergence, even if N is Banach. - If N is a Euclidean space, then unconditional convergence ≡ {\displaystyle \equiv } absolute-convergence.

  9. Riemann series theorem - Wikipedia

    en.wikipedia.org/wiki/Riemann_series_theorem

    In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges.