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 transformation that preserves the limits of convergent sequences.
[1] The linear sequence transformation can be applied to the divergent sequences of partial sums of divergent series to give those series generalized sums.
An infinite matrix
with complex-valued entries defines a regular matrix summability method if and only if it satisfies all of the following properties: An example is Cesàro summation, a matrix summability method with Let the aforementioned inifinite matrix
of complex elements satisfy the following conditions: and
be a sequence of complex numbers that converges to
lim
We denote
as the weighted sum sequence:
Then the following results hold: For the fixed
the complex sequences
approach zero if and only if the real-values sequences
approach zero respectively.
We also introduce
, for prematurely chosen
there exists
it's true, that
{\displaystyle {\begin{aligned}&\left|S_{n}\right|=\left|\sum _{m=1}^{n}\left(a_{n,m}z_{n}\right)\right|\leqslant \sum _{m=1}^{n}\left(\left|a_{n,m}\right|\cdot \left|z_{n}\right|\right)=\sum _{m=1}^{N_{\varepsilon }}\left(\left|a_{n,m}\right|\cdot \left|z_{n}\right|\right)+\sum _{m=N_{\varepsilon }}^{n}\left(\left|a_{n,m}\right|\cdot \left|z_{n}\right|\right)<\\& which means, that both sequences converge zero. lim Applying the already proven statement yields lim Finally, lim lim lim lim , which completes the proof.