real analysis – Interpretation of series solution

I have a function that is of the form

$f = a + b – cf$

I noticed that I can sove this in one of two ways. In the first way, the solution is

$f = frac{1}{1+c}(a+b)$

In the second way I can perform a recursion to get a series solution. There’s some intermediate steps to reach this realization, but in the end it has the form

$f = (a + b)*(1-c^2+c^4-c^6+ldots)$

I’ve checked and the series does in fact converge to the solution $1/(1+c)$ I presented prior. However, there is a slight difference. In the first solution, the domain for $c$ extends to infinity. In the second solution, the radius of convergence is 1, thus the domain for $c$ is bound. Here is the plot for a modest number of terms…

enter image description here

The convergence behavior is clear and everything to this point I’m good with. My question is centered around how to interpret the solution differences, specifically their domains. As shown in the figure, the domain for the first solution is infinite. Let’s assume nothing about the recursive approach was known (which lead to the series solution). Is the first solution actually wrong?

I’m trying to understand if one authors approach is superior to the other. Is there a benefit to using one approach vs the other? Within the radius of convergence they’re both the same in the limit of infinite terms, but beyond the radius of convergence, one solution goes one way while the other goes another way. The physical application does have the potential to take on values larger than $c=1$ due to what $c$ represents. The graph tells me that if $c=2$, go with the first solution approach, but I want to make sure I’m not missing something fundamental (e.g. there is an implicit assumption that $c<1$ in the first approach).