I need to say something about $\mathbf{R}$ - the radius of convergence of $\sum_0^\infty f(n)x^n$, where $f(x)$ is defined on $\mathbb{R}$ and has an asymptote $y = ax + b$, $x \rightarrow +\infty$. I also have three options:
i $\mathbf{R}$ can take no more than 2 different values.
ii $\mathbf{R} = 1$ or $\mathbf{R} \ge 2$
iii both i and ii are false
As I understand if $f(x)$ has an asymptote then $f(n) \approx an + b, n \rightarrow \infty$ then $lim_{n\rightarrow\infty}sup\sqrt[n]{|f(n)|} = lim_{n\rightarrow\infty}\sqrt[n]{|an + b|}$. Thus, the radius of convergence is equal to $\infty$ if $a = b = 0$, and is equal to $\frac{1}{lim_{n\rightarrow\infty}\sqrt[n]{|an + b|}} = 1$ otherwise. And according to my "solution" both i and ii are true, but in reality everything isn't so good, and the right answer is iii. I don't understand why. Can anybody help me, please?