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
members converges against the same limit as the original sequence, that is
with
implies
.
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. 33-36Harro 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
with
and a sequence of positive real numbers
with
one has
.
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
with
and
one has
.
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
and
such that
for all
. Due to
there exists a
with
for all
.
Now for all
the above yields:
:
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