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- Finding the fallacy in this broken proof 7 answers
This is the Grandi's series: $1-1+1-1+1-1+\dots$
The series can be equal to $0$
or to $1$
or to $1/2$
$$S=1-1+1-1+1-\dots,\quad\quad S=1-(1-1+1-1+1-1+1-...)$$ $$\Rightarrow S=1-S\Rightarrow 2S=1\Rightarrow S=1/2$$
Isn't this a contradiction? The integers are closed under addition and subtraction, but we get a fraction. Why?