in a geometric sequence, the generator is the number that one term is multiplied by to generate the next term.
The geometric series you describe converges. This can be proven with the ratio test.
The ratio test states that if the ratio of succeeding terms is a constant that is less than zero, the series converges.
lim(x→∞) Ax+1/Ax = r
If 0<r<1, then the series converges.
The exact value of a convergent, geometric series can often be found.
Let r be the ratio between two terms. For example, in the series 9, 3, 1, 1/3, 1/9, the ratio is 1/3.
For a starting number A, the sum of the sequence (S) is:
S = A + Ar + Ar2 + Ar3 + Ar4... [First equation]
If you multiply both sides by r, you get:
Sr =Ar + Ar2 + Ar3 + Ar4 + Ar5...
Now subtract this from the first equation:
S-Sr = A
All the other terms cancel out.
Solve for S:
S(1-r) = A
S = A / (1-r)
If your ratio is 1/2 and you start at 1, then the sum of the infinite series is S = 1 /(1-1/2) = 2
You can plug any starting number A and any ratio r into the equation to get the sum IF the series converges, which it will IF the ratio is between 0 and 1.