# Silverman-Toeplitz theorem

Let $\{a_{mn}\}$ be a double sequence of complex numbers and let $B$ be a positive real number such that:

1. 1.

$\sum_{n=0}^{\infty}|a_{mn}|\leq B$ for all $m=0,1,2,\ldots$

2. 2.

$\lim_{m\to\infty}\sum_{n=0}^{\infty}a_{mn}=1$

3. 3.

For every $n=0,1,2,\ldots$, it is the case that $\lim_{m\to\infty}a_{mn}=0$

Then, if the sequence $\{z_{n}\}$ converges, the series $\sum_{n=0}^{\infty}a_{mn}z_{n}$ converges and

 $\lim_{n\to\infty}z_{n}=\lim_{m\to\infty}\sum_{n=0}^{\infty}a_{mn}z_{n}$
Title Silverman-Toeplitz theorem SilvermanToeplitzTheorem 2013-03-22 14:51:28 2013-03-22 14:51:28 rspuzio (6075) rspuzio (6075) 10 rspuzio (6075) Theorem msc 40B05