I like your distinction between 'equal' and 'approach'.
These two words do not necessarily refer to the same concept.

YES, the series x=1/2+1/4+1/8+1/16+... EQUALS 1
because it satisfies the equation 2x=x+1
and the only solution of this equation is x=1.
So if you want the series to have any value at all
it had better be 1.

YES, the series 1/2+1/4+1/8+1/16+... APPROACHES 1
because the SEQUENCE
(1/2, 1/2+1/4, 1/2+1/4+1/8, 1/2+1/4+1/8+1/16,...)
= (1/2, 3/4, 7/8, 15/16, ...)
approaches 1.

Now consider another series: x=1+2+4+8+16+...

This series EQUALS -1
because it satisfies the equation 2x=x-1
and the only solution to this equation is x=-1.
So if you want the series to have any value at all
it had better be -1.

But the series does not APPROACH -1
because the sequence
(1, 1+2, 1+2+4, 1+2+4+8, ...)
=(1, 3, 7, 15,...)
does not approach -1.