I have the following recurrence relation for the coefficients $c_n$ of the power series: $y = \sum^\infty_{n=0}c_nx^n$
$$c_{n+1} = \frac{n+2}{5(n+1)}c_n$$
What is the radius of convergence of the power series?
I can find the radius of convergence by finding a formula for $c_n$ and then find the limit of $|\frac{c_n}{c_{n+1}}|$ as $n$ approaches $\infty$.
Is there a way to determine its radius of convergence without finding the formula first?
On
This may be a sledgehammer for your problem but there is a general theorem.
Poincare theorem$\color{blue}{{}^{[1]}}$
Given any recurrence relations with non-constant coefficients $$x_{n+k} + p_1(n)x_{n+k-1} + p_2(n)x_{n+k-2} + \cdots + p_k(n) x_{n} = 0\tag{*1}$$ such that there are real numbers $p_i, 1 \le i \le k$ with $$\lim_{n\to\infty} p_i(n) = p_i, \quad 1 \le i \le k$$ and the roots $\lambda_1, \lambda_2 \ldots, \lambda_k$ for the associated characteristic equation: $$\lambda^k + p_1 \lambda^{k-1} + \cdots + p_k = 0$$ have distinct moduli.
For any solution of $(*1)$, either $x_n = 0$ for all large $n$ or $\displaystyle\;\lim_{n\to\infty} \frac{x_{n+1}}{x_n} = \lambda_i\;$ for some $i$.
In short, if the coefficients of a recurrence relations converges and the corresponds $|\lambda_i|$ are distinct, then either the sequence $x_n$ terminates (i.e. infinite radius of convergence) or the radius of convergence is one of $\frac{1}{|\lambda_i|}$.
For your case, it is clear $c_n$ didn't terminate. Since the characteristic equation of your sequence "converge" to $\lambda - \frac15$, its radius of convergence is $5$.
Notes/References
You get $$\frac{5^{n+1}}{n+2}c_{n+1}=\frac{5^n}{n+1}c_n=…=c_0$$ so that you can get the explicit form of the power series, and from that read directly off its radius of convergence.
You can of course also directly evaluate the quotient expression $$\frac{c_n}{c_{n+1}}=\frac{5(n+1)}{n+2}.$$