Does $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{mn}^3=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}a_{mn}^3$ occur here?

Question

Let $\{a_{mn}\},~m,n \in \Bbb{N}$, be an arbitrary double sequence of real numbers then does the following equality occur always? $$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{mn}^3=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}a_{mn}^3$$

I think there is no reason for the equality of this iterated series in general but I cannot figure out how to construct a counter example. Can anyone please help me how to create a example of $\{a_{mn}\}$ such that the given equality does not hold...!!

Answer 1

The third power is just a distraction – if you have a doubly indexed sequence $b_{mn}$ for which the sums don't commute, you can obtain a counterexample for your equation by taking third roots, $a_{mn}=\sqrt[3]{b_{mn}}$. Can you find such a sequence $b_{mn}$?

Edit in response to the comment:

Examples abound. Take any convergent series $\sum_nc_n$ and any divergent series $\sum_nd_n$, and set

$$ b_{mn}=\begin{cases}d_m&n=1\;,\\-d_m&n=2\;,\\2^{-m}c_{n-2}&n\gt2\;.\end{cases} $$

Then $\sum_n\sum_m b_{mn}$ doesn't exist (since $\sum_mb_{m1}$ doesn't exist), whereas $\sum_m\sum_nb_{mn}=\sum_nc_n$.