Talk:Uniform convergence

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Thanks!Was a great help.... Stephan

[edit] Theorem assumption too strong?

In the section of theorems relating to the interchanging of limits, specifically the one pretaining to derivatives, I don't believe that uniform convergence of the original sequence of functions is necessary. It should be that if the sequence of functions {f_n(x)} converges pointwise and the sequence of derivatives {f_n'(x)} of the original sequence converges uniformly, then the original limit function f(x) = lim f_n(x) is differentiable and its derivative is equal to the limit of the sequence of the derivatives of the original sequence, f'(x) = lim f_n'(x). --anon

You are right. But I would prefer to leave it the way it is, it is more straightforward that way.
By the way, if the interval on which f_n are defined is bounded, it is enough for the sequence {f_n(x)} to be convergent just at one point. This together with the assumption that {f_n'(x)} converges uniformly, will imply that f_n converges uniformly to a function f and that f'(x) = lim f_n'(x). Oleg Alexandrov 02:37, 14 September 2005 (UTC)

[edit] Dirichlet or Seidel?

The article says:

Dirichlet then analyzed Cauchy's proof and found the mistake: the notion of pointwise convergence had to be replaced by uniform convergence.

On the other hand, infinite series says:

The theory of uniform convergence was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it successfully were Stokes and Seidel (1847-48).

Imre Lakatos (Proofs and Refutations, p. 135) also credits Stoke and Seidel, citing the following papers

  • P. L. Seidel (1847). "Note über eine Eigenschaft der Reihen, welche Discontinuirliche Functionen Darstellen". Abhandlungen der Mathmatisch-Physikalischen Klasse der Königlich Bayerischen Akademie der Wissenschaften 5: 381–393.
  • G. Stokes (1848). "On the Critical Values of the Sums of Peroidic Series". Transactions of the Cambridge Philosophical Society 8: 533–583.

Should this article be corrected? Gdr 22:26, 7 August 2006 (UTC)

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