Consider the power series $$\sum_{n=0}^{\infty}a_nz^n.$$
where, $a_0=0$ , $a_1=1$ , $a_n=a_{n-1}+a_{n-2}$.
Find the radius of convergence of the power series.
MY Attempt :
Clearly $\{a_n\}$ is a Fibonacci sequence.
Let, $R$ be the radius of convergence of the power series.
We have , $$\frac{1}{R}=\lim_n\sup\left|\frac{a_{n+1}}{a_n}\right|$$
$$=\lim_n\sup\left|\frac{a_n+a_{n-1}}{a_n}\right|$$
$$1+\lim_n\sup\left|\frac{a_{n-1}}{a_n}\right|.$$
But I can't write $\lim_n\sup\left|\frac{a_{n-1}}{a_n}\right|$ in terms of $R$ such that we can find out $R$ by solving the equation involving $R$.
Again we know that the $n$-th term of Fibonacci sequence is $$a_n=\frac{1}{\sqrt 5}\left[\left(\frac{1+\sqrt 5}{2}\right)^n-\left(\frac{1-\sqrt 5}{2}\right)^n\right].$$
From this I find that the radius of convergence of the power series is $\frac{2}{1+\sqrt 5}$.
Is this answer correct ?
If NOT what is the correct answer ?
But I want to find the radius of convergence NOT using the $n$-th term of Fibonacci sequence. How I can find it ?
Please help...
Thanks in Advance.........