-3

For example,

$1 + 2 + 3 + ... = -1/12$

or $2 + 4 + 8 + ... = -1$

What caused it to break? Are there some properties of real numbers that do this? Sorry if it's too vague because I was looking at group theory and there were things that didn't hold.

Another thing is why do some problems become ill-posed? For example, Wilkinson Polynomial, why do roots break by having too many multiplication of binomials?

user242559
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user29418
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    Nothing 1+2 + 3 + .... does not equal - 1/12 and 2+4+8 + .... does not equal -1. Math doesn't break. – fleablood May 16 '17 at 22:48
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    When we look at your first equation, the problem is that they are incorrect. The people who claim that that is true are using properties that do not apply at an infinity. It's not that math is "breaking," but more that a common usage of a property is applied to an improper area of math. – John Lou May 16 '17 at 22:48
  • LOL! (Face palm...) – PiE May 16 '17 at 23:26

2 Answers2

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The reason that your second example "breaks" is because the formula $$\frac{1}{1-r}$$ for the sum of an infinite geometric series only applies when $r \lt 1$. This is because the formula for the sum of the first $n$ terms of a geometric series with common ratio $r$ is $$\frac{1-r^{n+1}}{1-r}$$ If we want to find the sum of an infinite series, we must find $$\lim_{n \to \infty} \frac{1-r^{n+1}}{1-r}$$ When $r \lt 1$, then $r^{n+1}$ approaches $0$ and we are left with $$\frac{1}{1-r}$$ But when $r \gt 1$, then $r^{n+1}$ approaches $\infty$ and the formula can no longer be used.

Franklin Pezzuti Dyer
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Broadly, it comes about because it is often possible to apply some kind of manipulation to an expression that looks valid, but which is actually disallowed by something that is sneakily ignored. In the case of series like $1-1+1-1+\ldots$ and the like, the thing being ignored is that the proof only shows that if it has a value, then that value is the one you get from the algebra used. However, the fact that the series is divergent by even the simplest of tests shows that there isn't a meaningful value to the series, so the fact that we found out what that theoretical value is doesn't actually mean anything.

A similar principle applies to the various fake proofs that $0 = 1$ and the like - one of the steps typically introduces an additional solution (or removes an assumption) in a way that isn't immediately obvious. For example, starting with the assumption that $a = b$, then later multiplying something by $a - b$, or taking the square root of something and not considering the negative root.

ConMan
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