Etydhv

I am trying to find a sequence $x_pq$ such that $sum_p=1^inftysum_q=1^infty x_pq$ and $sum_q=1^inftysum_p=1^infty x_pq$ 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.

The iterated sums will be equal (if they converge) when all terms are nonnegative or, more generally, when we have absolute convergence.

Consider, on the other hand, $a_jk = 1/(k^2-j^2)$ if $jneq k$ and $a_jk = 0$ if $j=k$.

Here we have,

$$fracpi^28=tag*sum_j=1^infty sum_k=1\jneq k^infty frac1k^2 - j^2 neq sum_k=1^infty sum_j=1\j neq k^infty frac1k^2-j^2 = -fracpi^28 $$

To obtain the sum, note that

$$sum_k=1\jneq k^K frac1k^2-j^2 = frac12jsum_k=1\jneq k^K left(frac1k-j - frac1k+j right) \= frac12jleft(-sum_k=1^j-1frac1k + sum_k=1^K-jfrac1k - sum_k=j+1^K+jfrac1k + frac12j right)\= frac12jleft(frac1j +sum_k=j+1^K-jfrac1k - sum_k=j+1^K+jfrac1k + frac12j right)\ = frac12jleft(frac32j - sum_k=-j+1^j frac1K+kright) $$

and,

$$sum_k=1\jneq k^infty frac1k^2-j^2 = lim_K to inftyfrac12jleft(frac32j - sum_k=-n+1^n frac1K+kright) = frac34j^2$$

Thus,

$$sum_j=1^infty sum_k=1\jneq k^infty frac1k^2 - j^2 = sum_j=1^infty frac34j^2 = frac34 fracpi^26$$

Maybe you are looking for something like this: $$beginpmatrix1 &-1 & 0&0&0 & 0 &cdots \
0 & 1 & -1& 0 & 0 & 0&cdots\
0 & 0 & 1 & -1 & 0 & 0&cdots\
0 & 0 & 0 & 1 & -1 & 0&cdots \
0 & 0 & 0 & 0 & 1 & -1&cdots \
vdots &vdots &vdots & vdots & vdots & ddots&ddots
endpmatrix.$$

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 $$sum_p,q lvert x_p,qrvert$$ 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 $+infty$.