enow.com Web Search

Search results

  1. Results from the WOW.Com Content Network
  2. Harmonic series (mathematics) - Wikipedia

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

    The series = + = + + is known as the alternating harmonic series. It is conditionally convergent by the alternating series test , but not absolutely convergent . Its sum is the natural logarithm of 2 .

  3. Riemann series theorem - Wikipedia

    en.wikipedia.org/wiki/Riemann_series_theorem

    The alternating harmonic series is a classic example of a conditionally convergent series: = + is convergent, whereas = | + | = = is the ordinary harmonic series, which diverges. Although in standard presentation the alternating harmonic series converges to ln(2) , its terms can be arranged to converge to any number, or even to diverge.

  4. 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.

  5. 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.

  6. 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 ...

  7. Absolute convergence - Wikipedia

    en.wikipedia.org/wiki/Absolute_convergence

    Indeed, the sum of the absolute values of each term is + + + +, or the divergent harmonic series. According to the Riemann series theorem, any conditionally convergent series can be permuted so that its sum is any finite real number or so that it diverges. When an absolutely convergent series is rearranged, its sum is always preserved.

  8. Harmonic number - Wikipedia

    en.wikipedia.org/wiki/Harmonic_number

    The harmonic numbers roughly approximate the natural logarithm function [2]: 143 and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers.

  9. List of logarithmic identities - Wikipedia

    en.wikipedia.org/wiki/List_of_logarithmic_identities

    [9] [7] [10] As tends towards infinity, the difference between the harmonic numbers (+) and converges to a non-zero value. This persistent non-zero difference, ln ⁡ ( n + 1 ) {\displaystyle \ln(n+1)} , precludes the possibility of the harmonic series approaching a finite limit, thus providing a clear mathematical articulation of its divergence.