Hi, I saw this "proof" on some entertainment website: 0=0 0=0+0+0+0+0+0... 0=1-1+1-1+1-1+1-1+1-1... 0=1+0+0+0+0+0+0+0.... 0=1 We create something from nothing! Can someone explain what is wrong here? There has to be an illegal step!
The illegal step is that you're working with infinite sums. These are not at all well behaved. What you do is essentially the following: [tex](1-1) + (1-1) + (1-1) + ... = 1 + (-1+1) + (-1 + 1) + (-1 + 1) + ...[/tex] which is just rearranging the brackets. This is the illegal step. You can easily rearrange the brackets in finite sums ##a+(b+c) = (a+b)+c##, but not in infinite sums. You need to rigorously prove in infinite sums that you can do this. You can't in this case. You can in others.
I was expecting this. This shows a wrong attitude towards mathematics. Don't worry, you're in good company as most ancient mathematicians had this attitude. Your attitude is: I can do anything in mathematics as long as it looks right. For example, rearranging brackets looks right in the finite case, so it must work in the infinite case. This attitude has brought many paradoxes and contradictions to mathematics which took hundreds of years to resolve. The modern attitude is quite the opposite. It is: you can't do anything in mathematics without first showing rigorously that you can. So if we look at this question with the modern attitude, then we notice that we can rearrange brackets for finite sums. But we also immediately remark that doesn't imply anything for infinite sums. There is no reason to expect that a property like this holds if we have no rigorous proof of it. In fact, this example alone suffices to show that the property does not hold, it is a counterexample! This ancient versus modern attitude is very pervasive. For example, in ancient times it was thought that since stones fall towards the earth, that would imply that everything falls to the earth. Thus the earth is the center of the universe and everything falls towards it. The modern attitude is that there is no reason to expect things that hold for stones that they also hold for planets. It must be demonstrated somehow first. In fact, it has been demonstrated that it does not hold. The question you pose is a very difficult one to answer. This is why I give this lengthy response which doesn't really answer anything. The difficulty lies in what you expect to be the answer. How would you answer why something is not the case? If I were to ask you why the sun is not made of cheese, how could you possibly answer it? It is even not easy to answer why something is the case, let alone why something is not the case. So it is not clear to me at all what kind of answer you expect and what kind of answer would satisfy you. What satisfies me is to see that a counterexample exists and that the sum is infinite such that the reasoning of the finite sums do not apply. I guess that is not satisfactory for you, but I don't have a better answer than this. You must first think about this and try to formulate what kind of answer you expect to hear. Some things to consider: https://www.youtube.com/watch?v=36GT2zI8lVA
Just to point out how much of a counter example this is, let's see what would happen if one assumes that such a rearrangement is acceptable. Then surely we can also rearrange ##0=2+2-2+2-2\ldots## and ##0=3+3-3+3-3\ldots## and thus for any other number. Since ##0=1## and ##0## is the additive identity, ##1=1+0=1+1=2## which will also work with any choice of numbers! So in a mathematical world where arbitrary rearrangements of infinite series are acceptable, numbers really lose all meaning. It isn't just a weird conclusion that we choose to sweep under the rug. One either concludes that every number is equal and math is pointless, or this particular property does not hold for infinite sums.
You have to get analytical, and ask what do we mean by [itex]1-1+1-1+1-1+...[/itex]? As it stands, it's ambiguous. When dealing with infinite series, we define them as the limit of the partial sums of an infinite sequence of terms [itex]a_1,a_2,a_3...[/itex] The partial sums are [itex]S_n = \Sigma^n_{k=1}a_k=a_1+a_2+...+a_n[/itex]. The infinite sum is the limit [itex]\lim_{n \to \infty} S_n[/itex] (if it exists) which we write as [itex]\Sigma^{\infty}_{k=1}a_k[/itex]. This is what the infinite sum means. It only makes sense if the limit exists. In that case we say that the infinite sum converges. If it does not exist, then we say it diverges. So, what do we mean by [itex]1-1+1-1+...[/itex] ? Is it the infinite sum formed by the sequence of terms [itex]a_1 = 1, a_2 = -1, a_3 = 1, a_4 = -1, ...[/itex] ? In that case, the partial sums will be [itex]S_1 = 1, S_2 = 0, S_3 = 1,...[/itex] The sequence [itex]S_n[/itex] is alternating between 1 and 0 and will not converge. So the limit does not exist. However, by adding parentheses as such: [itex]1+(-1+1) + (-1+1) + (-1+1) + ...[/itex], and saying that [itex]a_1 = 1, a_2 = -1+1 = 0, a_3 = -1+1 = 0, ...[/itex] Then we are talking about an entirely different sequence of terms, and we get an entirely different sequence of partial sums [itex]S_n[/itex]. In this case, [itex]S_n = 1[/itex] for all n, and the limit is 1. The infinite sums are not the same, because the sequence we used to define what the sum meant is not the same. As to your question on why we can't rearrange the terms as such, what needs to be understood is that by adding the parantheses we are not talking about the same sum. There are circumstances where we may rearrange the sum as such and end up with the same result, but those are not satisfied here. In fact, rather view this example as a proof of that rearranging in such a way does not produce the same result in general.
In layman terms: infinite series is a sum of all terms in an infinite sequence. Such a sum can converge - that is, the more terms you add, the closer you get to some number. For example if you add all terms [tex]\frac 1 {2^1} + \frac 1 {2^2} + \frac 1 {2^3} + \frac 1 {2^4} ...[/tex] you will get 1. You can easily check that the sum gets closer and closer to 1 using just a calculator. But not every series converges. 1+1+1+1... doesn't - it just gets larger and larger. 1-1+1-1... is one of those series that don't converge, although it does it in a different way - it doesn't just grow infinitely, it oscillates between 1 and 0.