TITLE: Series that evaluates to different values
QUESTION [9 upvotes]: I am trying to find a sequence $x_{pq}$ such that $\sum_{p=1}^\infty\sum_{q=1}^\infty x_{pq}$ and $\sum_{q=1}^\infty\sum_{p=1}^\infty x_{pq}$ both exist and evaluate to different values. I am thinking of this as a matrix, but I can't seem to see how adding all the elements of it column wise vs row-wise would change the total sum? Any hints/examples would be helpful.

REPLY [7 votes]: Maybe you are looking for something like this: $$\begin{pmatrix}1 &-1 & 0&0&0 & 0 &\cdots \\
0 & 1 & -1& 0 & 0 & 0&\cdots\\
0 & 0 & 1 & -1 & 0 & 0&\cdots\\
0 & 0 & 0 & 1 & -1 & 0&\cdots \\
0 & 0 & 0 & 0 & 1 & -1&\cdots \\
\vdots &\vdots  &\vdots  & \vdots & \vdots & \ddots&\ddots 
\end{pmatrix}.$$
So that $x_{p,p} = 1$, $x_{p,p+1}=-1$ and $x_{p,q} = 0$ if $q \neq p,p+1$. 
Here we have $$\sum_{q=1}^{\infty} x_{p,q} = 0$$ for all $p \in \mathbb N$, and so $$\sum^\infty_{p=1} \sum^\infty_{q=1} x_{p,q} = 0.$$ However, we also have $$\sum_{p=1}^{\infty} x_{p,q} = 0 \,\,\,\, \text{ for } \,\,\,\, q \ge 2, \,\,\,\,\, \text{ but } \,\,\,\,\, \sum_{p=1}^{\infty} x_{p,1} = 1$$ and so $$\sum^\infty_{q=1} \sum^\infty_{p=1} x_{p,q} = 1.$$
As was said in the other answer, if $$\sum_{p,q} \lvert x_{p,q}\rvert$$ converges (which isn't the case here), then Fubini's theorem tells us we can sum in either dimension first and get the same answer. Similarly, if all $x_{p,q} >0$, we can do the same, by Tonelli's theorem and this allows for the possibility that the sums evaluate to $+\infty$.