Convergence of iterated series

Question

Suppose that we have a convergent iterated series $$ \sum_{i=1}^\infty\sum_{j=1}^\infty x_{i,j}, $$ where $\{x_{i,j}:i\ge1,j\ge1\}$ is a double sequence of real numbers (not necessarily non-negative). This iterated series doesn't converge absolutely. Does the iterated series $$ \sum_{j=1}^\infty\sum_{i=1}^\infty x_{i,j} $$ converge?

I've seen a few examples where both of the iterated series converges, but the limits are different. So I'm looking for an example where the first iterated series converges, but the second iterated series diverges.

Any help is much appreciated!

Answer 1

Consider the sequence given by the infinite matrix $(x_{ij})$,

$$x = \begin{pmatrix}1 \\ -1 & 2 \\ &-2 & 3 \\ &&-3 & 4\\ &&&\ddots \end{pmatrix}$$ where omitted entries are 0. Doing a vertical sum first gives $0+0+0+… = 0$. A horizontal sum first gives $1+1+1+1+… = ∞$.

If $\infty$ isn't what you call divergent,

$$x = \begin{pmatrix}1 \\ -1 & -2 \\ &2 & 3 \\ &&-3 & -4\\ &&&\ddots \end{pmatrix}$$

Vertical sum first still gives 0, but now horizontal sum gives you $1-3+5-7+…$ which is divergent (and I believe, $(C,2)$-summable! )

In fact if you want one where a horizontal sum leaves you with something that is just $(C,1)$-summable, $$x = \begin{pmatrix}1 \\ -1 & -1 \\ &1 & 1 \\ &&-1 & -1\\ &&&\ddots \end{pmatrix}$$ will do.

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Just for the record:

If one of them converges, and it does so absolutely in the sense that (say) $∑_i∑_j |x_{i,j}| < ∞ $, then so will the other and to the same value. This is a special case of Fubini's theorem, using the counting measure.