Help debunk a proof that zero equals one (no division)?

Unlike the more common variant of proof that 0=1, this does not use division.

So, the reasoning goes like this:

\begin{align}

0 &= 0 + 0 + 0 + \ldots && \text{not too controversial} \\
&= (1-1) + (1-1) + (1-1) + \ldots && \text{by algebra}\\
&= 1 + (-1 + 1) + (-1 + 1) \ldots && \text{by associative property}\\
&= 1\\
\\
&\therefore 0 =1
\end{align}

I can't help but feel that something went wrong here, specifically with the use of the associative property. However, I can't come up with a mathematically compelling reason.

Where's the error?

Answer

The error is that the "..." denotes an infinite sum, and such a thing does not exist in the algebraic sense. The usual way to make sense of adding infinitely many numbers is to use the notion of an infinite series: We define the sum of an infinite series to be the limit of the partial sums. (So the notion of convergence from analysis is involved in addition to algebra.)

Not all algebraic rules generalize to infinite series in the way that one might hope. When they fail, it is because something fails to converge. In this case, what fails to converge is the series that should appear between the two lines in the middle of the "proof":
$$1-1+1-1+1 \cdots.$$
Indeed, this series fails to converge because the
sequence of partial sums $\{1, 1-1, 1-1+1,\ldots\}$ oscillates between $1$ and $0$ and does not converge to any value.