# General Case of Convergence of a Power Series

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 $r=x^4$.
• I don't really see that. Writing out the first few terms where our c's are 1,2,3,4. I get $1 + 2x + 3x^2 + 4x^3 + x^4 2x^5 3x^6 + 4x^7 + \cdots$ I can't really see what the $r$ would be in this geometric series because the constants are changing and don't seem to have any relation. The $r$ would have to be $x*K$ ($K$ is some constant) Nov 17, 2014 at 5:57
• For any fixed $x$, it is the series $a+ar+ar^2+\cdots$ where $a=1+2x+3x^2+4x^3$ and $r=x^4$. If $|x|\lt 1$ the sum is $a/(1-r)$. Nov 17, 2014 at 8:03