Actual Infinity in Reality?

[GrandizerII]

1/2 + 1/4 + 1/8 +...<1     It is never equal to 1.    If you where to reach 1, then you would stop your infinite chain, and it would not be infinite.

1/2 + 1/2 = 1

1/2 + 1/4 + 1/4 = 1

1/2 + 1/4 + 1/8 + 1/8 = 1

If you hypothetically keep going with this, then you should conclude that:

1/2 + 1/4 + 1/8 + ... = 1

If not, then something must have changed the equation, and yet the left hand side of the equation has always been the same value and 1 is always equal to 1. So where's the change in value, pray tell?