>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.

The series 1/2+1/4+1/8+1/16+... equals 1 because it is convergent and satisfies 2x=x+1.

>So if you want the series to have any value at all
>it had better be 1.

Being convergent, the series does have a sum. It's not a matter of wanting or not wanting it.
>
>YES, the series 1/2+1/4+1/8+1/16+... APPROACHES 1
>because the SEQUENCE

The series does not approach 1, because from the above, it is just a number. The sequence below does approach 1.

>(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.

Series (possibly) equal, sequences (possibly) approach.

>Now consider another series: x=1+2+4+8+16+...
>
>This series EQUALS -1

No, it does not. The series is divergent and can't be assigned a value in any meaningful sense.

>because it satisfies the equation 2x=x-1

Consider y = 1 - 1 + 1 - 1 + 1 - ...

Then y = (1 - 1) + (1 - 1) + ... = 0,
but also y = 1 - (1 - 1) - (1 - 1) - ... = 1,
and also y = 1 - y, so that t = 1/2.

It's more difficult to manipulate 1+2+4+8+16+..., but the principle is the same. The series is divergent and could not be assigned any value.
>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.