Say we have the infinite series $$\sum_{n=0}^\infty a_nx^n.$$ We want to attempt to find an asymptotic function/relation to the above series. How does one begin? I have never dealt with this area of math (undergraduate at the moment). Does one need the closed form of the infinite series in order to find the asymptotic behavior?
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$\begingroup$ Asymptotic in what sense? If you're interested in the case when $x \to 0$ then you can just truncate the series at any point. If you're interested in the case when $x \to R^-$, where $R$ is the radius of convergence of the series, then you may find something useful in my answer here. $\endgroup$ – Antonio Vargas Mar 27 '14 at 21:47
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$\begingroup$ @AntonioVargas I'm pretty unfamiliar with this topic but I know for example you can find the asymptotic behavior of $n!$ (Stirling's Approximation). I want to approximate my infinite series. I want to see how fast it converges. $\endgroup$ – H5159 Mar 27 '14 at 21:51
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$\begingroup$ Ok, it sounds like you're interested in seeing how fast the tail $\sum_{n=N}^{\infty} a_n x^n$ tends to zero as $N$ tends to infinity. $\endgroup$ – Antonio Vargas Mar 27 '14 at 21:53
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$\begingroup$ Right, I have the closed form for my series. Is there any way using that or my series to find how fast? Some direction would be helpful. I don't know where to begin. $\endgroup$ – H5159 Mar 27 '14 at 21:54
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$\begingroup$ If you have a closed form for your series then you can probably take advantage of Taylor's theorem. For example, en.wikipedia.org/wiki/… $\endgroup$ – Antonio Vargas Mar 27 '14 at 21:57
