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  2. Limit of a sequence - Wikipedia

    en.wikipedia.org/wiki/Limit_of_a_sequence

    The plot of a convergent sequence {a n} is shown in blue. Here, one can see that the sequence is converging to the limit 0 as n increases. In the real numbers, a number is the limit of the sequence (), if the numbers in the sequence become closer and closer to , and not to any other number.

  3. 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.

  4. Kolmogorov's three-series theorem - Wikipedia

    en.wikipedia.org/wiki/Kolmogorov's_three-series...

    This is a special case of a more general result from martingale theory with summands equal to the increments of a martingale sequence and the same conditions ([] =; the series of the variances is converging; and the summands are bounded). [2] [3] [4]

  5. Modes of convergence - Wikipedia

    en.wikipedia.org/wiki/Modes_of_convergence

    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 subsequence of the partial sums of the original series.

  6. Alternating series - Wikipedia

    en.wikipedia.org/wiki/Alternating_series

    Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms a n converge to 0 monotonically, but this condition is not necessary for convergence.

  7. Convergence proof techniques - Wikipedia

    en.wikipedia.org/wiki/Convergence_proof_techniques

    Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity.. There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence.

  8. Series (mathematics) - Wikipedia

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

    Series with sequences of partial sums that converge to a value but whose terms could be rearranged to a form a series with partial sums that converge to some other value are called conditionally convergent series. Those that converge to the same value regardless of rearrangement are called unconditionally convergent series.

  9. List of integer sequences - Wikipedia

    en.wikipedia.org/wiki/List_of_integer_sequences

    0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, ... "subtract if possible, otherwise add" : a (0) = 0; for n > 0, a ( n ) = a ( n − 1) − n if that number is positive and not already in the sequence, otherwise a ( n ) = a ( n − 1) + n , whether or not that number is already in the sequence.