Find a sequence where $\sum_{n\ge1}\sum_{m\ge1}{a_{nm}}=0$ and $\sum_{m\ge1}\sum_{n\ge1}{a_{nm}}=\infty$. Find a sequence where $\displaystyle\sum_{n\ge1}\sum_{m\ge1}{a_{nm}}=0$ and $\displaystyle\sum_{m\ge1}\sum_{n\ge1}{a_{nm}}=\infty$.
How do I find such a sequence? Can you give me a hint so that I can find it?
I thought of thinking this as a partial sum $S_n$ and then $\lim S_n=0$ if so, $S_n=\frac{1}{n}$.
 A: An example:
$$
\begin{matrix}
 1 & -1 & 0 & 0 & \cdots \\
 0 & 2 & -2 & 0 & 0 & \cdots \\
 0 & 0 & 3 & -3 & 0 & 0 & \cdots \\
 0 & 0 & 0 & 4 & -4 & 0 & 0 & \cdots 
\end{matrix}
$$
All row partial sums are eventually equal to zero, and all column partial sums are eventually equal to one.
Note that any example must involve negative numbers: If any of the iterated series $\sum_{n\ge1}\sum_{m\ge1}{a_{nm}}$ or $\sum_{m\ge1}\sum_{n\ge1}{a_{nm}}$ converges absolutely then the other converges absolutely as well, and their values are equal.

For more general examples we can choose two arbitrary sequences $(b_n)$ and $(c_n)$, and define
$$
\begin{align}
 a_{n,n} &= b_n + \sum_{k=1}^{n-1}(b_k -c_k) \\
 a_{n,n+1} &= \sum_{k=1}^{n}(c_k -b_k) \\
 a_{n, m} & = 0 \text{ otherwise.}
\end{align}
$$
The first three rows look like this:
$$
\begin{matrix}
 b_1 & c_1-b_1 & 0 & 0 & \cdots \\
 0 & b_1+b_2-c_1 & c_1+c_2-b_1-b_2 & 0 & 0 & \cdots \\
 0 & 0 & b_1+b_2+b_3-c_1-c_2 & c_1+c_2+c_3-b_1-b_2-b_3 & 0 & 0 & \cdots 
\end{matrix}
$$
Then
$$
\sum_{n\ge1}\sum_{m\ge1}{a_{nm}} = \sum_{n\ge1} c_n
$$
and
$$
\sum_{m\ge1}\sum_{n\ge1}{a_{nm}} = \sum_{m\ge1} b_m
$$
which shows that we can achieve arbitrary values for the two iterated series. The above example corresponds to $b_n=1$ and $c_n = 0$.
