Tannery's Theorem: Definition & Use

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In summary, Tannery's Theorem is a mathematical theorem that discusses the convergence of an infinite series. It has been referenced in various sources, including a mathematical journal article and a problem set from a university math course. While it may be difficult to find information on this subject, it appears to be a simple theorem with conditions that allow for the interchange of an infinite sum with a limit.
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What is Tannery's Theorem? I can't seem to find a definition of what it is. And how do you use it?
 
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Yes, it is hard to find anything on this subject, but

According to one source (http://www.wpr.aaugonline.net/gazette/ ), E.H.Neville discussed Tannery's theorem in The Mathematical Gazette, July 1930 Issue, p. 166.

Apparently it is a thereom concerning convergence of an infinite series.

See also a reference to Tannery's theorem (see §63, page 161, Advanced Calculus by G.A. Gibson, MacMillan 1954; also the February 2002, Volume 109, 196-200 AMM article by Josef Hofbauer) from http://www.numbertheory.org/papers.html

and there is a problem set from a math course at DePaul University - http://condor.depaul.edu/~rjohnson/foma/exercises.pdf - which refers to Tannery's theorem. This refers to another source: Johnsonbaugh/Pfaffenberger: Foundations of Mathematical Analysis.

Good luck.
 
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Just a note, Johnsonbaugh/Pfaffenberger book doesn't actually have any info on Tannery's theorem, it appears that exercise set is just supplementary problems though it does ask you to prove Tannery's theorem. It looks simple enough to prove, and gives some conditions that will allow you to interchange an infinite sum with a limit.
 

FAQ: Tannery's Theorem: Definition & Use

What is Tannery's Theorem?

Tannery's Theorem, also known as the Tannery-Steinhardt Theorem, is a mathematical theorem that relates to the convergence of series. It states that if a series of real or complex numbers satisfies certain conditions, then the series must either converge absolutely or diverge.

Who is Tannery and when was the theorem created?

Jules Tannery was a French mathematician who developed this theorem in the late 19th century. He published it in his paper "Sur la convergence des séries", which was presented to the French Academy of Sciences in 1893.

What are the conditions for Tannery's Theorem to apply?

The conditions for Tannery's Theorem are as follows:

  • The series must be a series of real or complex numbers.
  • The series must have non-negative terms.
  • The terms of the series must decrease monotonically.
  • The limit of the ratio of consecutive terms must exist and be finite.

What is the use of Tannery's Theorem?

Tannery's Theorem is useful in determining the convergence or divergence of series in mathematics. It can be used to simplify the process of determining the convergence of a series by eliminating the need to use other convergence tests.

Can Tannery's Theorem be applied to all series?

No, Tannery's Theorem can only be applied to series that meet the specific conditions outlined in the theorem. If a series does not meet these conditions, then Tannery's Theorem cannot be used to determine its convergence or divergence.

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