Given the sequence:
$$a_n = \frac{2n+1}{2n-3}a_{n-1} -\frac{1}{2n-3}$$
Find its radius of convergence.
HINT: Show that the coefficients have a lower limit and that they do not grow faster than $5^n$
What I have tried:
First I looked for a pattern; we know that $a_{n-1}$ is:
$$a_{n-1} = \frac{2(n-1)+1}{2(n-1)-3}a_{n-2} -\frac{1}{2(n-1)-3} = \frac{2n-1}{2n-5}a_{n-2} -\frac{1}{2n-5}$$
So we get:
$$a_n = \frac{\Pi_{j=1}^{n}(2j+1)}{\Pi_{i=1}^{n}(2i-3)}a_{0} -\frac{1}{\Pi_{i=1}^{n}(2i-3)} = (2n+1)(2n-1)a_0 -\frac{1}{\Pi_{i=1}^{n}(2i-3)} $$
I was trying to apply the ratio test but things get messy.
I was thinking about the hint; I know I'll have to argue about lim sup $\lvert a_{n+1} \frac{1}{a_{n}}\rvert \le 5$ but I do not get to this point.
Answer: The radius of convergence is 1.