I've run into some asymptotic analysis in research here and there and largely it feels pretty magical to me. My research has led me to consider the following question which I haven't the slightest idea how to address:
If we have two series $f(x) = \sum_{n=0}^{\infty} a_n x^n$ and $g(x) = \sum_{n=0}^{\infty} b_n x^n$ with $a_n,b_n\in\Bbb R$ which converge everywhere and $a_n \sim \frac{b_n}{n}$, can we conclude anything about the asymptotic behavior of $f$ in relation to $g$?
For example: if, say, $g(x) \sim x^{-\frac{1}{2}}$ at infinity, could we conclude that $f$ decays at least as fast as $x^{-\frac{1}{2}}$ at infinity? It seems reasonable to me to think that this would be the case but asymptotic analysis is pretty counter-intuitive at times, especially when the coefficients are allowed to oscillate.