Do we have $\sum_{n=1}^\infty 0=0$? Simple question: Do we have
$$\sum_{n=1}^\infty 0=0$$
?
Mathematically this seems obvious, but in practice I am very uncomfortable with this. Because nothing is perfect, so $0$ might not be quite zero, say $0.0000000000001$. Then,
$$\sum_{n=1}^\infty 0.0000000000001=\frac{1}{10^{13}}\sum_{n=1}^\infty 1=\infty.$$
 A: We do, in fact, have $\displaystyle\sum_{n=1}^\infty 0=0$. We also have $\displaystyle\sum_{n=1}^\infty 0.0000000000001=\infty$.
In fact, to make this more general, we have:
$$\sum_{n=1}^\infty x=
\begin{cases} \infty&x>0\\
0&x=0\\
-\infty&x<0\end{cases}$$
This is a discontinuous function. (Note that, strictly speaking, this is not a function, since $\infty$ isn't a number.)
I see where your confusion comes from. For one thing, infinite sums can be counterintuitive. For another, discontinuous functions don't happen a lot in nature. We expect things to change gradually as $x$ changes, rather than having a huge jump like we have above.
In any case, the value of an infinite sum, by definition, depends on its finite sums. In fact, we have:
$$\sum_{n=1}^\infty 0=\lim_{N\to\infty}\left(\sum_{n=1}^N0\right)=\lim_{N\to\infty}0=0$$
(If you haven't seen the $\lim$ notation before: Roughly, it means "see what happens as $N$ gets larger and larger." There is a more rigorous, but more complicated, definition.)
The main reason that the infinite sum is $0$, is that every partial sum adds up to $0$. Thus, almost by definition, the infinite sum adds up to $0$, too.
And, remember: When dealing with infinities, discontinuities aren't uncommon.
