Double Series Let $u_{m,n}$ be a number determinate for all positive integral values of m and n; consider the array $u_{1,1},u_{1,2},u_{1,3},\ldots$
$u_{2,1},u_{2,2},u_{2,3},\ldots$
$u_{3,1},u_{3,2},u_{3,3},\ldots$
Let the sum of the terms inside the rectangle, formed by the first $m$ rows of the first $n$ columns of this array of terms, be denoted by $S_{m,n}$.
If a number $S$ exists such that, given any arbitrary positive number $\epsilon$, it is possible to find integers $m$ and $n$ such that $\left| S_{\mu ,v} -S\right| < \epsilon $ whenever both $\mu > m$ and $v>n$, we say that the double series of which the general element is $u_{\mu ,v}$ converges to the sum $S$,and we write $\lim _{\mu\rightarrow \infty ,v\rightarrow \infty }S_{\mu,v}=S$. This definition is practically due to Cauchy.
If the double series, of which the general element is $\left|u_{\mu,v}\right|$, is convergent, we say that the given double series is absolutely convergent.
Since $u_{\mu,v} = \left(S_{\mu,v} - S_{\mu,v-1}\right)\left(S_{\mu - 1,v} - S_{\mu -1,v -1}\right)$, it is easily seen that, if the double series is convergent, then $\lim _{\mu\rightarrow \infty ,v\rightarrow \infty }u_{\mu ,v}=0.$
Stolz'necessary and sufficient condition for convergence (But first proven by Pringsheim). A condition for convergence which is obviously necessary is that, given $\epsilon$, we can find m and n such that $\left|S_{\mu + \rho,v+\sigma} - S_{\mu,v-1}\right| < \epsilon$ whenever $\mu > m$ and $v>n$ and $\rho,\sigma$ may take any of the values 0,1,2,....
The condition is also sufficient; for, suppose it satisfied; then, when $\mu>m +n$,$\left|S_{\mu + \rho,v+\rho} - S_{\mu,\mu}\right| < \epsilon$. Therefore, $S_{\mu,\mu}$ has a limit $S$; then, making $\rho$ and $\sigma$ tend to infinity in such a way that $\mu + \rho = v + \sigma$, we see that $\left|S - S_{\mu,v}\right|\leq \epsilon$ whenever $\mu > m$ and $v >n$; that is to say, the double series converges.
An absolutely convergent double series is convergent. For if the double series converges absolutely and if $t_{m,n}$ be the sum of $m$ rows and $n$ columns of the series of moduli, then,given $\epsilon$, we can find $\mu$ such that, when $p>m>\mu$ and $q>n>\mu$,$t_{p,q}-t_{m,n}<\epsilon$, but $\left| S_{p,q}-S_{m,n}\right| \leq t_{p,q}-t_{m,n}$ and so $\left| S_{p,q}-S_{m,n}\right| < \epsilon $ when $p>m>\mu, q>n>\mu$; and this is the condition that the double series should converge.
Methods of Summing double Series Let us suppose that $\sum _{v=1}^{\infty }U_{\mu,v}$ converges to the sum $S_{\mu}$. Then $\sum _{\mu=1}^{\infty }S_{\mu}$ is called the sum by rows of the double series; that is to say, the sum by rows is $\sum _{\mu =1}^{\infty }\left( \sum _{v=1}^{\infty }u_{\mu },v\right) $. Similarly, the sum by columns is defined as $\sum _{v=1}^{\infty }\left( \sum _{\mu =1}^{\infty }u_{\mu },v\right) $. That these two sums are not necessarily the same is shown by the example $S_{\mu ,v}=\dfrac {\mu - v } {\mu+v}$, in which the sum by rows is -1, the sum by columns is +1; and S does not exist.
Pringsheim's Theorem: If $S$ exists and the sums by rows and sum by columns exist; then each of these sums is equal to S. For since $S$ exists, then we can find $m$ such that $\left| S_{\mu ,v} -S\right| < \epsilon $ whenever both $\mu > m$ and $v>m$. And therefore, since $\lim _{v\rightarrow \infty }S_{\mu ,v}$ exists, $\left| \left( \lim _{v\rightarrow \infty }S_{\mu ,v}\right) -S\right| \leq \epsilon$; that is to say, $\left| \sum _{p=1}^{\mu }S_{p}-S\right| \leq \epsilon $ when $\mu>m$ and so the sum by rows converges to $S$. In a like manner the sum of columns converge to $S$.