I try to find a limit of a series of a side from the rectangle of a golden ratio (Golden Cut, Goldener Schnitt). Just for fun to learn more about series. After hours of thinking I could need some help to find a method to determine the limit analytically.
Here is what I have done so far:
Sketch: 
Goal: The goal is to find the limit of the series based on the sequence for side a or b.
Sequence for side a: $$ a_0 = a $$ $$ a_1 = b_0 = b $$ $$ a_2 = b_1 = \frac{b^2}{a} $$ $$ a_3 = b_2 = \frac{b^4}{a^3} $$ $$ a_n = b_{n-1} = \frac{b^{2n-2}}{a^{2n-3}}$$
Sequence for side b: $$ b_0 = b $$ $$ b_1 = \frac{b_0}{a_0}b_0 = \frac{b_0^2}{a_0} = \frac{b^2}{a} $$ $$ b_2 = \frac{b_1}{a_1}b_1 = \frac{b_1^2}{a_1} = \frac{b^4}{a^3} $$ $$ b_3 = \frac{b_2}{a_2}b_2 = \frac{b_2^2}{a_2} = \frac{b^6}{a^5} $$ $$ b_n = \frac{(b_{n-1})^2}{a_{n-1}} = \frac{b^{2n}}{a^{2n-1}} $$
Now I try to make each sequence independent from the other. For that I introduce the constant factor c as a ratio of site b and a. $$ c = \frac{b}{a} $$
under the condition
$$ 0 < c < 1 $$
The sequence for site a is:
$$ a_0 = c^0 a = a $$ $$ a_1 = c^1 a = c a$$ $$ a_2 = c^2 a $$ $$ a_3 = c^4 a $$ $$ a_4 = c^6 a $$ $$ a_n = c^{2n-2} a$$
The sequence for site b is:
$$ b_0 = c^1 a = c a $$ $$ b_1 = c^2 a $$ $$ b_2 = c^4 a $$ $$ b_n = c^{2n} a $$
At this point, I determine the limit of the sequences. $$ \lim \limits_{n \to \infty} (a_n) = \lim \limits_{n \to \infty} c^{2n-2} = 0 $$ $$ \lim \limits_{n \to \infty} (b_n) = \lim \limits_{n \to \infty} c^{2n} = 0 $$
And then the determination of the limit of the series: $$ \lim \limits_{n \to \infty} \sum_{i=0}^n a_i = ??? $$ $$ \lim \limits_{n \to \infty} \sum_{i=0}^n b_i = ??? $$
Now I'm at a point where I don't know what to do. The method of the Epsilon-Neighborhood does not work, because for this I must have at least one guess for a limit. Cauchy criteria as well. I tried to reorder the partial sum of the sequences without success.
I know that it is difficult to find a analytic solution. Maybe it is not possible in this case. But maybe I'm missing a methodological approach that I really need to know about. What would you do at this point?
Thank you for your time!