Let me explain, the summation can either be 0 or a 1 given a finite number of periods n that denote the number of +1s and -1s, where if n is the total number of periods, then there are i +1s and n-i -1s. This means that at any point in time of the infinite series your solution can be 0 or 1.

Your reasoning of being 1/2 is simplified and is actually the mean of the result, which makes sense since the arithmetic mean of 0 and 1 is 1/2.

Now consider this. There are two equations you mention (1-1)+(1-1)+(1-1)+… = 0+0+0+0+, which would represent infiinite summation of (1-1)n, which will always equal 0.

and infinite sum of 1+(1-1) = 1+infinite sum (1-1) = always is 1.

The problem with your logic comes from the fact that you first mention that we have an equal number of +1 and -1, but then in your proof you present the possibility that the number of -1 and +1 are not equal in number which means i != n-i. However, no sequence or series can determine if you get a -1 or a +1, which means probability must be involved, and if there is an equal chance of getting a -1 or a +1 on the very next cycle, then your expected sum will = 1/2 by stochastic properties.

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