I have dealt with radius of convergence for simple series, but this one is literally complex:
$\frac{1}{1-z-z^2}=\sum_{n=0}^\infty c_nz^n$
How does one calculate the radius of convergence here? I can't just use the ratio test? Any ideas?
What methods would I use in general? I haven't much experience with complex analysis
Answer
As Micheal pointed, the radius of convergence is just the distance from the origin of the closest singularity, so $\rho=\frac{\sqrt{5}-1}{2}$. You can achieve that also by noticing that:
$$ c_n = F_{n+1} = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right]\tag{1}$$
since $(1)$ holds for $n\in\{0,1\}$ and $(1-x-x^2)\cdot\frac{1}{1-x-x^2}=1$ implies:
$$ \forall n\geq 0,\quad c_{n+2}=c_{n+1}+c_n.\tag{2}$$
No comments:
Post a Comment