Cauchy's limit theorem, named after the French mathematician Augustin-Louis Cauchy, describes a property of converging sequences. It states that for a converging sequence the sequence of the arithmetic means of its first
n
(an)
an\toa
(a1+ … +an)/n \toa
If the arithmetic means in Cauchy's limit theorem are replaced by weighted arithmetic means those converge as well. More precisely for sequence
(an)
an\toa
(pn)
1 | |
p1+ … +pn |
\to0
p1a1+ … +pnan | |
p1+ … +pn |
\toa
This result can be used to derive the Stolz–Cesàro theorem, a more general result of which Cauchy's limit theorem is a special case.[2]
For the geometric means of a sequence a similar result exists. That is for a sequence
(an)
an>0
an\toa
\sqrt[n]{a1 ⋅ a2 ⋅ … ⋅ an} \toa
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 allows for assigning real values to certain divergent series and leads to the concept of Cesàro summation and summable series. In this context Cauchy's limit theorem can be generalised into the Silverman–Toeplitz theorem.[1] [4]
Let
\varepsilon>0
N\in\N
|ak-a|\leq\tfrac{\varepsilon}{2}
k\geqN
\limn
1 | |
n |
N | |
\sum | |
k=1 |
(ak-a)=0
M\in\N
\left| | 1 |
n |
N | |
\sum | |
k=1 |
(ak-a)\right|\leq
\varepsilon | |
2 |
n\geqM
Now for all
n\geqmax(N,M)
\begin{align} \left| | 1 |
n |
n | |
\left(\sum | |
k=1 |
ak\right)-a\right|&=\left|
1 | |
n |
n | |
\sum | |
k=1 |
(ak-a)\right|=\left|
1 | |
n |
N | |
\sum | |
k=1 |
(ak-a)+
1 | |
n |
n | |
\sum | |
k=N+1 |
(ak-a)\right|\ &\leq\left|
1 | |
n |
N | |
\sum | |
k=1 |
(ak-a)\right|+
1 | |
n |
n | |
\sum | |
k=N+1 |
|ak-a|\leq
\varepsilon | |
2 |
+
(n-N)\varepsilon | |
2n |
\leq\varepsilon. \end{align}