why the two ways of adding elements of array yield different results? Given the array, $$\begin{matrix}
-1 & 0 & 0 & 0 & \ldots \\
1\over2 & -1 & 0 & 0 & \ldots \\
1\over4 & 1\over2 & -1 & 0 & \ldots \\
1\over8 & 1\over4 & 1\over2 & -1 & \ldots \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\end{matrix}$$ Here $\sum_i\sum_ja_{ij} = -2$, while $\sum_j\sum_ia_{ij} = 0$. Why both sums are different? 

I assume you think that the two should be the same because you are adding all the same elements, simply ordering them differently. While it is true that rearranging an absolutely convergent series will give you the same result, no matter how you order the elements $a_{ij}$ in your double series (in order to make it a single series) you will have a conditionally convergent series, which can be rearranged to converge to any real number and even diverge. This fact is known as the Riemann rearrangement theorem, and is very cool.
Edit: As Robert Israel pointed out, you can't actually make this into a conditionally convergent single series as the limit of the terms can't go to $0$ (since there are infinitely many $-1$'s). So in this case the failure of rearrangement preserving the value of the series is even stronger.
