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. 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

Formal statement

Let the aforementioned inifinite matrix of complex elements satisfy the following conditions: and z_{n} be a sequence of complex numbers that converges to. We denote S_{n} as the weighted sum sequence:. Then the following results hold:

Proof

Proving 1.

For the fixed the complex sequences z_{n}, S_{n} and a_{i, j} approach zero if and only if the real-values sequences, and approach zero respectively. We also introduce. Since, for prematurely chosen there exists , so for every we have. Next, for some it's true, that for every and. Therefore, for every which means, that both sequences and S_{n} converge zero.

Proving 2.

. Applying the already proven statement yields. Finally, , which completes the proof.

Citations