Checking my understanding: $1 - 1 + 1 - 1 + 1 - ... = \frac{1}{2}$

If you have some method of attaching a sum to a series (e.g. Cesaro summation, as discussed in Ayesha's answer, or Abel summation) which is linear (i.e. the limit of a linear combination of two series coincides with the same linear combination of the limits), and if $1- 1 + 1 - 1 + \cdots$ is summable with respect to this method, then this argument shows that the value of the sum will have to be $1/2$.

By itself, this argument won't tell you whether a given summation method applies to your series, though.

The series of course doesn't converge - that doesn't mean it doesn't have a sum. Indeed, note that the Cesaro means of the series tend to 1/2, and thus the Cesaro sum is 1/2.

Consider the partial sums of the series $s_n$. Then we say the (C, 1) sum of the series is the limit $$\lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^{n}s_k $$ if this limit exists.

For your series here, we have the partial sums $1, 0, 1. . .$ and thus the Cesaro means are $1, \frac{1}{2}, \frac{2}{3}. . . $ a sequence tends to 1/2. Your proof is a heuristic demonstration of this rigorous fact.