An example of this is the summation of 1+2+3...=-1/12. By some reason, you cannot change the digits of that to 1+(1+1)+(1+1+1)... which would be equal to -1/2. -1/12 is not equal to -1/2 though.
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1By what rule do you have $1+1+ \cdots = -1/2$ and $1 + 2 + 3 + \cdots = -1/12$? Sure, some rules exist that yield these results (various regularizations of infinite series), but in order for us to give you an answer, we need to know what framework you're working in. – Ben Grossmann Jul 02 '15 at 23:03
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Presumably, associativity generally fails for non-convergent series. As a classic example, we can show (by placing parentheses) that $$ 1-1+1-1+1- \cdots $$ is equal to both $0$ and $1$. – Ben Grossmann Jul 02 '15 at 23:06
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possible duplicate of Does the equality $1+2+3+.... = -\frac{1}{12}$ lead to a contradiction? – Jonas Meyer Jul 03 '15 at 01:50
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In "standard" mathematics, such as that taught in first-year calculus or analysis classes, the infinite series $1+2+3+\cdots$ and $1+1+1\cdots$, as well as others such as $1-1+1-1+1-\cdots$ have no value at all. That is because if we say that one of those series has a value, doing things like grouping terms together can give a different value. That is basically what you are pointing out. There are some groupings of terms that makes $1+2+3+\cdots=-\frac1{12}$ and $1+1+1+\cdots=-\frac 12$, but other groupings give other answers.
This kind of thing led to mathematicians of the early 1800's to come up with formal definitions of when an infinite series has a value and when it does not. They looked at the kinds of series that gave multiple answers and those that did not. They came up with the current standard definition: If the partial sums of the first $n$ terms comes to a finite limit then we say that limit is the value of the infinite series. Otherwise the series has no value at all.
That stopped most of the paradoxes, but a few still remained. They saw that even some "good" series might give the same value for all groupings of terms but would change value for different arrangements (orders) of terms. They came up with further classifications of series, saying that "conditionally convergent" series could not be rearranged, but "absolutely convergent" series could since rearranging would give the same answer.
So "standard" mathematics says that you are playing with nonsensical things and are therefore getting nonsensical results. There are non-standard ways of looking at these series, but they vary in exactly what they do. That is the reason for @Omnomnomnom's first comment that we need to know which framework you are working in.
I left out many details in this short discussion. Ask if you need more on any point.
Rory Daulton
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