Interchange of summation order

Remarks

By the nonnegative case of Fubini's theorem for series, the hypothesis says that the family $(a_{m,n})$ over $\N \times \N$ is summable. The same theorem, applied once with rows first and once with columns first, shows that both iterated sums equal the sum of the family. The hypothesis cannot be dropped. Let $a_{m,m} = 1$, $a_{m,m+1} = -1$ for all $m$, and $a_{m,n} = 0$ otherwise. Each row sums to $0$, so $\sum_m \sum_n a_{m,n} = 0$. Column $0$ sums to $1$ and every other column sums to $1 - 1 = 0$, so $\sum_n \sum_m a_{m,n} = 1$. Example of use: with the Riemann zeta function, $\sum_{k \ge 2} (\zeta(k) - 1) = \sum_{k \ge 2} \sum_{n \ge 2} n^{-k}$. Swapping the nonnegative sums gives $\sum_{n \ge 2} \sum_{k \ge 2} n^{-k} = \sum_{n \ge 2} \frac{1}{n(n-1)} = 1$.

Going deeper

For a student

The result is used to swap two sums, which is one of the most common steps in analysis and number theory. To apply it, bound the double sum of absolute values; for nonnegative terms the swap is always allowed, possibly with value $+\infty$. The counterexample above shows why conditionally convergent double sums need care.