Why might 1+1–1+1–1…=1/2? (And why it doesn’t)

Background

Grandi’s series is the infinite sum 1-1+1-1+1-1+1–1…, or to put into words the summation of the alternating sequence of 1’s and -1’s. The terms of this sequence are always 1 or -1 and so it doesn’t converge to any single value. If we take any finite portion of this series, the value of the summation will be either 1 or 0 depending on if the finite sequence is of the form (1–1)+(1-1)+...+1 (sum of 1), or of the form (1-1)+(1-1)+...+(1-1) (sum of 0). We might also want to think about the sum of the whole infinite series. Since the last term determines if the sum equals 1 or 0 and since infinite series do not have a last term, the infinite sum ought to be undefined. In spite of this, there is a convincing argument that the sum should be 1/2.

The Convincing Argument

A1: let S = 1-1+1...
A2: let S' = 1 - S = 1 - (1-1+1...) = 1-1+1-1... = S
A3: So we can say that S' = 1 - S = S
A4: 1 = S + S = 2 * S
A5: 1/2 = S

Above are the steps of the proof that Grandi’s series takes the value of 1/2. First we define the value of Grandi’s series as S, we then define a new series S' = 1-S. From here we show that S = S', which allows us to then say that 2*S = 1, and that S = 1/2, that is that Grandi’s sum S, has a value of 0.5 (the average of the 2 possible outcomes).

Refusing to be Convinced

When I first saw this I was a little unsettled. I mainly could not believe that the addition operator (which closes integers under addition), when applied to a sequence with all-integer terms, could yield a value outside of the set of integers. This prompted me to look at the claim more closely. While watching this video I noticed a subtle assumption that is made in steps A3 and A4 in the above proof. I have described this below:

Claim: 1 = S + S
Derived from fact: 1 - S = S (true)
When seeing this, we are used to moving -S from the LHS to RHS
But we are actually doing this: 1 - S + S = S + S
For this to simplify to: 1 = S + S (the claim)...
...we assume that -S + S (a.k.a: S - S) = 0
Is this true? Not necessarily.

The point is that although we can usually say that X - X = 0 for any X, since in this case X is the sum of a divergent infinite series, the value here is indeterminate, meaning this statement does not necessarily hold. Because of the indeterminate value of S (1 or 0), the expression: S - S may just as easily evaluate to 1 – 0 = 1, 0 – 1 = -1, 0 - 0 = 0, or 1 - 1 = 0 (used in step B4 below). Following this logic through, we get the following:

B1: 1 - S = S
B2: 1 - S + S = S + S
B3: 1 + (S - S) = S + S
B4: 1 + {-1,0,1} = S + S
B5: {0, 1, 2} = S + S = 2 * S
B6: {0, 1/2, 1} = S

From this we do get that S = 1/2 is a possible simplification of the equation, but that it is not the only value of S. That is, S is still indeterminate, but one of the possible values it may collapse to is 1/2. From this we also get a more complete picture and can see that S = 1, and S = 0 are also valid simplifications.

So S = 1/2 is not the unique solution, but it’s still valid? — Not quite.

There is still a problem with this! As I originally complained, S = 1/2 should not be valid at all since it violates the fact that integers are closed under addition. This is to say that where X and Y are integers, X + Y should always result in an integer. Since S is a sum of integers, it should not be able to evaluate to a non-integer and so 1/2 should not be a valid evaluation of S. The key to this is another basic assumption, that S + S = 2 * S is false. Remember that S is undefined and may take a value of 0 or 1. Knowing this we can see the issue as described below:

fact: X + X = 2 * X
assumption: X = X
claim: S + S = 2 * S
We can see that this claim is only true if S = S...
...whereas we know that this is false since S may be either 1 or 0.
This means that this fact is only true in the case that the summed S values are synced up which is not given to be true.
Conclusion: S + S != 2 * S

By disallowing S + S from being equated with 2 * S, we are no longer able to use division on our equation. Walking back to B5 (above), we get the following:

B5: {0, 1, 2} = S + S = 2 * S
B5': {0, 1, 2} = S + S (!= 2 * S)

Because we cannot introduce multiplication to this equation, we cannot further simplify using division, which prevents 1/2 from being a valid candidate for S to evaluate to.

Conclusion

Through writing this I aimed to showcase an interesting circumstance in which intuitive arithmetic assumptions do not hold, and how this can lead to us deriving unintuitive results from an intuitive process. Specifically, we took a closer look at Grandi’s sum, the claim that it evaluates to 1/2, and the reasons why Grandi’s sum should not evaluate to 1/2 in spite of a convincing proof otherwise.