I could express it like so:

The proposition starts with ( 1 - 1 ) + ( 1 - 1 ) + ( 1 - 1 ) = 0

Then expands it, but it does so ignoring the last part. We can make the same infinite expansion keeping the representation of the last part, like so:

( 1 - 1 ) + ( 1 - 1 ) + ... + ( 1 - 1 ) = 0 , instead of ( 1 - 1 ) + ( 1 - 1 ) + ... = 0

If we take the (-1) from the new representation, we can see that in reality what we get is:

1 - (1 - 1 + 1 - 1 + ... + 1 - 1 + 1) = 0

while the 1 - (1 - 1 + 1 - 1 + ... ) = 0 notation makes it look like if the series ends, in the infinite, like

1 - (1 - 1 + 1 - 1 + ... + 1 - 1 ) = 0

Which is not the true form of the first proposed series.