Complex power series : Radius of Convergence Could anyone please suggest me how to deal with these questions (Complex Variable) :
Note that all problems are in $\mathbb{C}$.
3.1 Determine the radius of convergence $\rho$ of each of the following series :
$$\sum_{n=0}^\infty \frac{z^n}{n!}, \sum_{n=0}^\infty \frac{z^n}{n^2}, \sum_{n=0}^\infty \frac{z^n}{n}, \sum_{n=0}^\infty n!z^n $$
I use the ratio test and obtain the result : $\infty, 1, 1, 0$. , respectively.
The problem is : 3.2 For those with $\rho = \infty$, can you conclude anything about "convergence at infinity" ? I do not know what is "convergence at infinity". I search both in the text books and the internet already, but it seems no clues. 
3.3 Find the radius of convergence of the power series :
$$\sum a_nz^n,$$
where $a_0 = 0, a_1 = 1, a_n= a_{n-1} + a_{n-2}$ for all $n > 1$.
I do not know how to handle this series since $a_n$ is defined recursively, so most of method for determining radius of convergence cannot apply.
 A: I think the solution is provided by Pp.. in his comments but if you didn't get it then first follow this link  from Wikipedia which is about this series and its generating function. The second hint as provided by Pp.. is that you need to find the singularities of the generating function of this power series which is $f(z) = \frac{1}{1-z-z^2}.$ As is obvious the singularities are $\frac{\sqrt{5} \pm 1}{2}$. The minimum of which is $\frac{\sqrt{5} - 1}{2}$ and that is the radius of convergence.
PS: The other way to find the radius of convergence is through continued fraction you can go for that as well but for that you need to know what continued fraction are.
EDIT: Please look into this link.
Hope it helps.
A: I simplify it into this form :
Let $P_N(z) = \sum_{n=1}^N a_nz^n$ for all $N \in \mathbb{N}$. 
Then let $$Q_N(z) = (z^2+z-1)P_N(z) = (z^2+z-1) \sum_{n=1}^N a_nz^n \\ 
= \sum_{n=0}^N a_nz^{n+2} + \sum_{n=0}^N a_nz^{n+1} - \sum_{n=0}^N a_nz^n \\
= -a_0 + (a_0 - a_1)z + (a_0 + a_1 - a_2)z^2 + (a_1 + a_2 - a_3)z^3 + ... + (a_{N-2} + a_{N-1} - a_N)z^N + a_Nz^{N+1} + a_{N-1}z^{N+1} + a_Nz^{N+2} \\
= -z + a_{N+1}z^{N+1} + a_Nz^{N+2}.$$
Any suggestion to go next ?
