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Yes, they are equal.jonjacson said:Why? Why must I look one by one? Why can't I just put two terms together?
I mean, let's forget series, Isn't this +1-1 equal to (+1-1)?
Suppose we allow -1+1-1+1... to be a valid infinite summation with a single answer.
On one hand, we have -1 +(+1-1) + (+1-1) +(+1-1) + ... = -1 +0 +0+0...= -1
On the other hand we have (-1+1) + (-1+1) + (-1+1) + ... = 0+0+0... = 0
Which one is correct? This violates the associative property of addition, which is fundamental to all mathematics.
In math, a summation has a valid total only if changing the order or clustering of terms does not change the number that the partial sums eventually converge to. The associative and commutative properties of addition are not violated by the math definition of an infinite sum.
We either have to give up on -1+1-1+1... having a well-defined total, or we have to give up the associative and commutative properties of addition. The choice is clear. You will see eventually that "summations" like -1+1-1+1... are not very interesting and we haven't given up much.