The Question
If $f(x) = \sum c_nx^n$, where $c_{n+4} = c_n$ for all $n\ge 0$, find the interval of convergence of the series and a formula for $f(x)$
My Work and Question
I haven't been able to do much. It's very clear that the values of $c_0,c_1,c_2,c_3$ are very important. If they're all $0$ than the function is obviously convergent, but that is only one case of many. If any of them is not $0$ then our interval of convergence must be $(-1,1)$. Else our function would diverge to positive or negative infinity. Not really sure where to go from here. I think my interval must be correct because anything in that range is going to gradually approach $0$ but I'm not really sure how I could find the function. Any hints to get me started would be greatly appreciated.
Hint: In the interesting case where the coefficients are not all $0$, we have a geometric series.
Added: Fix $x$, with $|x|\lt 1$. Then our series is $a+ar+ar^2+ar^3+\cdots$, where $a=c_0+c_1x+c_2x^2+c_3x^3$ and $r=x^4$.