<s>[INST] Why is there different results when calculating a double sum? Consider:
$$
\begin{matrix}
-1 & 0 & 0 & 0& 0& \ldots\\
1/2 & -1 & 0 & 0& 0& \ldots\\
1/4 & 1/2 & -1 & 0& 0& \ldots\\
1/8 & 1/4 & 1/2 & -1& 0& \ldots\\
1/16&1/8 & 1/4 & 1/2 & -1& \ldots\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots
\end{matrix}
$$
When I calculate the sum of each column first, I got $0,0,0\ldots$ then sum them up the answer should be $0$. But when I calculate the sum of each row, I got $-1, -1/2, -1/4, -1/8,\ldots$, then the sum will be $-2$. Why is there different results when I calculate the sum?
 [/INST] The sum is not absolutely convergent (http://en.wikipedia.org/wiki/Convergent_series) so we can make the sum equal whatever we want.
With divergent series the way that we order the series may change the sum. As @DuAravis said the series can be written as
\begin{align*}
S_n &=(-1) +(\frac{1}{2} -1) +(\frac{1}{4}+\frac{1}{2} -1)+(\frac{1}{8}+\frac{1}{4}+\frac{1}{2} -1) +\ldots=-2 \quad (1)
\end{align*}
or
\begin{align*}
S_n &=(-1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8} +\ldots) +\ldots (0-1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8} +\ldots) +\ldots=0 \quad (2)
\end{align*}
but we can also rearrange the matrix as follows,
$$
\begin{matrix}
-1 & 0 & 0 & 0& 0& \ldots\\
1/2 & -1 & 0 & 0& 0& \ldots\\
1/2 & 1/4 & -1 & 0& 0& \ldots\\
1/2 & 1/4 & 1/8 & -1& 0& \ldots\\
1/2 &1/4 & 1/8 & 1/16 & -1& \ldots\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots
\end{matrix}
$$
now if we sum the columns we get $\infty$. 
Also, we need to be careful about multiplying. For example, if multiply each term by $x$. Then by (2) this shouldn't change the total sum,
\begin{align*}
S_n &=x \times (-1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8} +\ldots) +\ldots x \times (0-1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8} +\ldots) +\ldots=x \times 0=0
\end{align*}
but now if we use (1),
\begin{align*}
S_n &=x \times (-1) +x \times (\frac{1}{2} -1) +x \times (\frac{1}{4}+\frac{1}{2} -1)+(\frac{1}{8}+\frac{1}{4}+\frac{1}{2} -1) +\ldots=-2x \quad (1)
\end{align*}
The problem I think is that you're working with $\infty$'s so when you manipulate the sum it's like adding, subtracting, or multiplying by $\infty$. Here are two famous examples http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF and http://en.wikipedia.org/wiki/Grandi%27s_series.
Here are some links on mathexchange Sum of divergent series and Sum of infinite divergent series
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