Summability theory | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-31

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Divergent_series


00:02:43 1 History
00:04:18 2 Examples
00:04:31 3 Theorems on methods for summing divergent series
00:04:46 4 Properties of summation methods
00:05:05 5 Classical summation methods
00:05:50 5.1 Absolute convergence
00:06:05 5.2 Sum of a series
00:06:19 6 Nørlund means
00:06:35 6.1 Cesàro summation
00:06:48 7 Abelian means
00:08:58 7.1 Abel summation
00:10:05 7.2 Lindelöf summation
00:10:31 8 Analytic continuation
00:11:13 8.1 Analytic continuation of power series
00:11:34 8.2 Euler summation
00:15:15 8.3 Analytic continuation of Dirichlet series
00:15:55 8.4 Zeta function regularization
00:16:30 9 Integral function means
00:16:59 9.1 Borel summation
00:19:11 9.2 Valiron's method
00:19:57 10 Moment methods
00:20:21 10.1 Borel summation
00:20:31 11 Miscellaneous methods
00:22:31 11.1 Hausdorff transformations
00:24:32 11.2 Hölder summation
00:24:43 11.3 Hutton's method
00:24:53 11.4 Ingham summability
00:26:16 11.5 Lambert summability
00:26:27 11.6 Le Roy summation
00:26:38 11.7 Mittag-Leffler summation
00:26:47 11.8 Ramanujan summation
00:26:58 11.9 Riemann summability
00:27:17 11.10 Riesz means
00:27:36 11.11 Vallée-Poussin summability
00:28:29 12 See also
00:28:32 13 Notes
00:29:44 14 References
00:29:54 Zeta function regularization
00:30:56 −1, then its value at s
00:31:39 −1 is −1/12, assigning a value to the divergent series 1 + 2 + 3 + 4 + .... Other values of s can also be used to assign values for the divergent sums ζ(0)
NaN:NaN:NaN −1/2, ζ(−2)
NaN:NaN:NaN Integral function means
NaN:NaN:NaN r. In this case one defines the sum as above, except taking the limit as x tends to r rather than infinity.
NaN:NaN:NaN Borel summation
NaN:NaN:NaN In the special case when J(x)
NaN:NaN:NaN Valiron's method
NaN:NaN:NaN e−G(n), and a0 + ... + ah is to be interpreted as 0 when h 0.
NaN:NaN:NaN Moment methods
NaN:NaN:NaN Borel summation
NaN:NaN:NaN e−x dx for positive x and 0 for negative x then μn
NaN:NaN:NaN Miscellaneous methods
NaN:NaN:NaN Hausdorff transformations
NaN:NaN:NaN Hölder summation
NaN:NaN:NaN Hutton's method
NaN:NaN:NaN Ingham summability
NaN:NaN:NaN Lambert summability
NaN:NaN:NaN Le Roy summation
NaN:NaN:NaN Mittag-Leffler summation
NaN:NaN:NaN Ramanujan summation
NaN:NaN:NaN Riemann summability
NaN:NaN:NaN Riesz means
NaN:NaN:NaN Vallée-Poussin summability
NaN:NaN:NaN See also



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SUMMARY
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For an elementary calculus-based introduction, see Divergent series on Wikiversity.
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series

  
    
      
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    {\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n}}.}
  The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme.
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