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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, $$\sum_{n=1}^\infty 0.0000000000001=\frac{1}{10^{13}}\sum_{n=1}^\infty 1=\infty.$$

user232646
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  • Then this is completely different sums. – m0nhawk Apr 19 '15 at 13:03
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    Zero is always zero. The partial sums are all zero. Thus the limit is zero. – Gregory Grant Apr 19 '15 at 13:03
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    Zero is always zero even in numerical form (float, double, int, etc.). – thang Apr 19 '15 at 13:05
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    One of the charms of mathematics is that things can be perfect. – Simon S Apr 19 '15 at 13:11
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    Fantastic proof. I think we could modify it to show that no equation in mathematics is correct. – Martin Brandenburg Apr 19 '15 at 13:15
  • Why do you doubt that the zero in the summand might not be zero? – GFauxPas Apr 19 '15 at 13:21
  • Could we say that infinite summation is not continuous? – wlad Apr 19 '15 at 13:23
  • So small changes in the function being summed can cause big changes in the output. – wlad Apr 19 '15 at 13:24
  • The summand is a constant, a constant function if you wish, and constants don't change. That has nothing to do with continuity. I can post an $\epsilon - M$ proof as an answer, if you'd like. – GFauxPas Apr 19 '15 at 13:31
  • You're expecting everything to be a continuous function — your intuition tells you that, if $x$ changes gradually, $y$ will change gradually, too. (Or, more specific to your objection: You expect that if you know $x$ approximately, you know $y$ approximately, too.) This only happens with continuous functions. There are discontinuous functions, too. – Akiva Weinberger Apr 19 '15 at 13:38

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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.

Akiva Weinberger
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