Double series

Remarks

Example: for $a_{m,n} = 2^{-m} 3^{-n}$ one has $S_{M,N} = \left(\sum_{m \le M} 2^{-m}\right)\left(\sum_{n \le N} 3^{-n}\right)$. The two factors tend to $2$ and $\tfrac32$, so the double series converges to $3$. Non-example: for $a_{m,n} = 1$ when $m = n$ and $0$ otherwise, $S_{M,N} = \min(M, N) + 1$, which is unbounded, so the double series diverges. A convergent double series can have unbounded terms. With $a_{0,n} = n$, $a_{1,n} = -n$ and $a_{m,n} = 0$ for $m \ge 2$, one has $S_{M,N} = 0$ for all $M \ge 1$. So it converges to $0$, although its rows do not form convergent series.