The Grandi Series (1-1+1-1+1 . . .) was actually proved as follows:
Z=1-1+1-1+1 . . .
1-Z=1-(1-1+1-1+1 . . .)
1-Z=1-1+1-1+1-1 . . .
1-Z=Z => 2Z=1 => Z=1/2

This is perfectly true and can be used without controversy in the remaining half of the proof. Adding Z to both sides would work as an alternative proof, but it would have to be done as follows:
Z=1-1+1-1+1 . . .
Z+Z=1-1+1-1+1 . . .
+1-1+1-1 . . .
After which everything cancels out, except the 1:
2Z=1 => Z=1/2.

The way that was shown in the article wouldn't work because there is no way to know whether the second Grandi Series is being started at +1 or -1 in the original Grandi Series.

## Grandi Series and Mistake in the Article

The Grandi Series (1-1+1-1+1 . . .) was actually proved as follows:

Z=1-1+1-1+1 . . .

1-Z=1-(1-1+1-1+1 . . .)

1-Z=1-1+1-1+1-1 . . .

1-Z=Z => 2Z=1 => Z=1/2

This is perfectly true and can be used without controversy in the remaining half of the proof. Adding Z to both sides would work as an alternative proof, but it would have to be done as follows:

Z=1-1+1-1+1 . . .

Z+Z=1-1+1-1+1 . . .

+1-1+1-1 . . .

After which everything cancels out, except the 1:

2Z=1 => Z=1/2.

The way that was shown in the article wouldn't work because there is no way to know whether the second Grandi Series is being started at +1 or -1 in the original Grandi Series.