The magnificence of Infinity!

Let the equation equate to a variable $S$ .

$1 - 1 + 1 - 1 + 1 - 1 + ... = S$

From here, we can bracket the equations differently:

$(1 - 1) + (1 - 1) + (1 - 1) + ... = S$

$1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ... = S$

We can then sum the equations, solving every sub-equation within the brackets:

$1 + 0 + 0 + 0 + 0 + 0 + 0 + ... = 2S$

$S = \frac{1}{2}$

This answer, however, may be wrong as by bracketing the equations differently, we can derive $0 = 1$ . I remember reading somewhere you take partial sums of the equation, then average between the answers, which leads to the answer being $\frac{1}{2}$ due to the partial equations not converge on a single value but rather jumping between 0 and 1 constantly.