Let's put an end to this.

1/2 + 1/4 +1/8 .. is a decaying geometric series with first term 1/2 and common ratio 1/2.

Using the formula for the sum of a decaying geometric series:

S = first term /

= 1/2 /

= 1/2 / 1/2

= 1

There we go - the series equals 1