Cauchy's Limit Theorem
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Cauchy's limit theorem, named after the French mathematician
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...
, describes a property of converging sequences. It states that for a converging sequence the sequence of the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...
s of its first n members converges against the same limit as the original sequence, that is (a_n) with a_n\to a implies (a_1+\cdots+a_n) / n \ \to a.
Konrad Knopp Konrad Hermann Theodor Knopp (22 July 1882 – 20 April 1957) was a German mathematician who worked on generalized limits and complex functions. Family and education Knopp was born in 1882 in Berlin to Paul Knopp (1845–1904), a businessman i ...
: ''Infinite Sequences and Series''. Dover, 1956, pp. 33-36
Harro Heuser Harro Heuser (December 26, 1927 in Nastätten – February 21, 2011 in Bingen am Rhein, Bingen) was a Germans, German mathematician. In German-speaking countries he is best known for his popular two-volume introduction into real Mathematical analys ...
: ''Lehrbuch der Analysis – Teil 1'', 17th edition, Vieweg + Teubner 2009, ISBN 9783834807779, pp
176-179
(German)
The theorem was found by Cauchy in 1821, subsequently a number of related and generalized results were published, in particular by
Otto Stolz Otto Stolz (3 July 1842 – 23 November 1905) was an Austrian mathematician noted for his work on mathematical analysis and infinitesimals. Born in Hall in Tirol, he studied at the University of Innsbruck from 1860 and the University of Vienna fr ...
(1885) and
Ernesto Cesàro Ernesto Cesàro (12 March 1859 – 12 September 1906) was an Italian mathematician who worked in the field of differential geometry. He wrote a book, ''Lezioni di geometria intrinseca'' (Naples, 1890), on this topic, in which he also describes ...
(1888).


Related results and generalizations

If the arithmetic means in Cauchy's limit theorem are replaced by
weighted arithmetic mean The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. Th ...
s those converge as well. More precisely for sequence (a_n) with a_n\to a and a sequence of positive real numbers (p_n) with \frac \to 0 one has \frac\to a . This result can be used to derive the
Stolz–Cesàro theorem In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. It is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time. The Stolz–Cesàro theorem can b ...
, a more general result of which Cauchy's limit theorem is a special case. For the
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
s of a sequence a similar result exists. That is for a sequence (a_n) with a_n>0 and a_n\to a one has \sqrt \ \to a. The arithmetic means in Cauchy's limit theorem are also called Cesàro means. While Cauchy's limit theorem implies that for a convergent series its Cesàro means converge as well, the converse is not true. That is the Cesàro means may converge while the original sequence does not. Applying the latter fact on the partial sums of a
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
allows for assigning real values to certain divergent series and leads to the concept of
Cesàro summation In mathematical analysis, Cesàro summation (also known as the Cesàro mean or Cesàro limit) assigns values to some Series (mathematics), infinite sums that are Divergent series, not necessarily convergent in the usual sense. The Cesàro sum ...
and summable series. In this context Cauchy's limit theorem can be generalised into the
Silverman–Toeplitz theorem In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in series summability theory characterizing matrix summability methods that are regular. A regular matrix summability method is a linear sequence transfor ...
.Johann Boos: ''Classical and Modern Methods in Summability''. Oxford University Press, 2000, ISBN 9780198501657, p. 9


Proof

Let \varepsilon>0 and N \in \N such that , a_k - a, \leq \tfrac for all k \geq N. Due to \lim_ \frac \sum_^N (a_k - a) = 0 there exists a M \in \N with \left, \frac \sum_^N (a_k - a)\ \leq \frac for all n \geq M . Now for all n \geq \max(N,M) the above yields: :\begin \left, \frac \left(\sum_^n a_k\right) - a\ & = \left, \frac \sum_^n (a_k - a)\ = \left, \frac \sum_^N (a_k - a) + \frac \sum_^n (a_k - a)\ \\ & \leq \left, \frac \sum_^N (a_k - a)\ + \frac \sum_^n , a_k - a, \leq \frac + \frac \leq \varepsilon. \end{align}


References


Further reading

* Sen-Ming: ''Note on Cauchy's Limit Theorem''. In: ''The American Mathematical Monthly'', Vol. 57, No. 1 (Jan., 1950), pp. 28–31
JSTOR


External links



at SOS math
''Cesàro Mean''
- proof of Cauchy's limit theoren at the ProofWiki Theorems about real number sequences Convergence tests