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  2. Geometric series - Wikipedia

    en.wikipedia.org/wiki/Geometric_series

    The convergence of a geometric series can be described depending on the value of a common ratio, see § Convergence of the series and its proof. Grandi's series is an example of a divergent series that can be expressed as 1 − 1 + 1 − 1 + ⋯ {\displaystyle 1-1+1-1+\cdots } , where the initial term is 1 {\displaystyle 1} and the common ratio ...

  3. Rate of convergence - Wikipedia

    en.wikipedia.org/wiki/Rate_of_convergence

    The staggered geometric progression () =,,,,, …, / ⌊ ⌋, …, using the floor function ⌊ ⌋ that gives the largest integer that is less than or equal to , converges R-linearly to 0 with rate 1/2, but it does not converge Q-linearly; see the second plot of the figure below. The defining Q-linear convergence limits do not exist for this ...

  4. Limit of a sequence - Wikipedia

    en.wikipedia.org/wiki/Limit_of_a_sequence

    A sequence that does not converge is said to be divergent. [3] ... (terminus) of a geometric series in his work Opus Geometricum (1647): ...

  5. Convergent series - Wikipedia

    en.wikipedia.org/wiki/Convergent_series

    The ratio test and the root test are both based on comparison with a geometric series, and as such they work in similar situations. In fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse, however, is not true.

  6. Series (mathematics) - Wikipedia

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

    An example of a convergent series is the geometric series ... any series of real numbers or complex numbers that converges but does not converge absolutely is ...

  7. Convergence tests - Wikipedia

    en.wikipedia.org/wiki/Convergence_tests

    In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, ... is a geometric series with ratio ...

  8. Geometric mean - Wikipedia

    en.wikipedia.org/wiki/Geometric_mean

    The sequences converge to a common limit, and the geometric mean is preserved: ... The fundamental property of the geometric mean, which does not hold for any other ...

  9. Uniform convergence - Wikipedia

    en.wikipedia.org/wiki/Uniform_convergence

    A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = ⁡ (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).