Suppose $f(z)=\displaystyle\sum_{n=0}^\infty a_nz^n$ has infinite radius of convergence (where the $a_n\in\mathbb{C}$). Is there any tests which can be done on the sequence $\{a_n\}$ to determine whether $f$ converges to zero along the real line? That is, whether $\displaystyle\lim_{x\to\pm\infty}f(x)=0$?

Necessary or sufficient conditions are welcome.

  • $\begingroup$ No universal test for all entire functions. Furthermore, even for such a special case as the polynomials of degree 5 or higher, there is no universal formula for locating the zeros (and, accordingly, testing whether they lie on the real line). This is nonexistence is the content of Abel's Theorem: maths.ed.ac.uk/~aar/papers/abel.pdf $\endgroup$ – avs Aug 24 '16 at 16:45
  • $\begingroup$ Do you mean "wether $f\color{red}{(z)}$ converges to 0 for all $z\in\mathbb C$"? $\endgroup$ – sranthrop Aug 24 '16 at 16:57
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    $\begingroup$ @avs I am familiar with Abel's Theorem, but I do not actually care about the location of the zeros, only the limit along the real line. $\endgroup$ – Trevor Richards Aug 24 '16 at 16:57
  • $\begingroup$ @sranthrop Thanks for the question, I updated to make it clear. $\endgroup$ – Trevor Richards Aug 24 '16 at 16:59
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    $\begingroup$ Of course any test for the question as currently stated would have to be sensitive to every coefficient, making it very unlike the ratio and root tests. A weakening would be, can we give criterion for $\{a_n\}$ such that some derivative of $f$ approaches zero along $\mathbb{R}$, or such that $f-p$ approaches zero along $\mathbb{R}$ for some polynomial $p$. $\endgroup$ – Trevor Richards Aug 24 '16 at 17:13

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