Given a function $f(x)$ and an asymptotic series $\phi_n(x)$ as $x\rightarrow x_0$. What is the easiest way to determine if the asymptotic expansion of $f(x)$ exists when using the series $\phi_n(x)$?
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This is a good question. I'm not sure what techniques exist for this, but I often use numerical tools to try to calculate the coefficients in such an expansion. This often leads me to conclusions like "the expansion seems to have only odd powers of $x$" or "there are probably some powers of logarithms mixed in since these limits don't seem to exist". I then follow up these methods with analytical approaches which could yield the types of expansions I expect. – Antonio Vargas Jul 28 '17 at 09:15