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Given a Taylor series expression of a function $f(x)=\sum_{n=0}^{\infty}a_nx^n$, with an infinite radius of convergence, are there any basic/standard techniques for understanding the asymptotic behavior of the function as $x\to\infty$? How is the asymptotic behavior of some of the more famous functions, e.g. Bessel functions, Hypergeometric functions, etc. determined?

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    A lot use their equivalent integral formulations and then use special techniques to compute approximations. Just check the wiki page on Bessel functions and you'll see it. – Gregory Sep 13 '22 at 20:18
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    There are wildly different functions with Taylor series with infinite convergence radius. For example, $;e^x, \sin x, \cos x, 1;$ All have infinite radius of convergence and they all behave differently when $;x\to\infty;$ , so I don't think much can be said... – DonAntonio Sep 13 '22 at 20:20
  • If you have the $a_n$ in any sort of recursive form, convert to a differential equation and then work with that. – eyeballfrog Sep 13 '22 at 20:23
  • The examples I'm familiar with involve rewriting the function as an integral and applying some form of Laplace's method, and I'd guess the Bessel functions can be done this way: https://en.wikipedia.org/wiki/Laplace%27s_method – Qiaochu Yuan Sep 13 '22 at 21:25
  • See a possible method here. You can also look up Hayman's method, the Harris-Schoenfeld method, Grosswald's method and Wayman's method. – Gary Sep 14 '22 at 08:20

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