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I have a problem about the plotting of a function which is defined as the power series $$F(\eta)= \left[1+\frac{10.75}{\eta^{15/4}}+O\left(\frac{1}{\eta^{15/2}}\right)\right]^{-7/4} \biggr[1 + \frac{6G_4}{G_2\eta^{15/4}} +\frac{15 G_6}{G_2\eta^{15/2}}+\frac{28 G_8}{G_2\eta^{45/4}}+\frac{45 G_{10}}{G_2\eta^{15}} + \frac{66G_{12}}{G_2\eta^{75/4}} + \frac{91G_{14}}{G_2 \eta^{45/2}}+ O\left(\frac{1}{\eta^{105/4}}\right)\biggr]^{-1}$$ where $G_i$ are some known constant coefficients and the series as a function of $\eta$ is well-defined in the limit of $\eta\to+\infty$. The problem is that when considering this series as a normal function of $\eta$ and trying to plot it along the entire axis of this variable, the resulting plot produced by Mathematica goes to zero as $\eta$ approaches $0$. While the previous series has an evident divergence for $\eta\to0$ as seen if we stop the expansion to some low order $$F(\eta)\approx 1-\left(\frac{75.25}{4}+\frac{6G_4}{G_2}\right)\frac{1}{\eta^{15/4}}+O\left(\frac{1}{\eta^{15/2}}\right)$$ In fact the plot that I got is enter image description here which I cannot really understand. For $\eta\to+\infty$ the function is correctly approaching $1$, but why for $\eta$ close to $3$ the function starts to decrease and it arrives to zero? Here is my code

G2 = -1.8452283;
G4 = 8.33410;
G6 = -95.1884;
G8 = 1458.21;
G10 = -25889;
G12 = 5.02*^5;
G14 = -1.04*^7;
F[x_] := ((1 + 10.75/(x^(15/4)))^(-7/4))*((1 + (182*G14)/(
  2*G2*x^(45/2)) + (132*G12)/(2*G2*x^(75/4)) + (56*G8)/(
  2*G2*x^(45/4)) + (30*G6)/(2*G2*x^(15/2)) + (12*G4)/(
  2*G2*x^(15/4)) + (90*G10)/(2*G2*x^15))^(-1))
Plot[F[x], {x, -3, 10}, PlotStyle -> ColorData[97][1]]
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    $\begingroup$ Please post actual, copyable, Mathematica code, not images of code! And for us to help, we probably need also to see in the code the values of all the G's. $\endgroup$
    – murray
    Mar 2, 2019 at 21:30
  • $\begingroup$ Can you maybe give a reference to the book/paper where this divergent series came from? $\endgroup$
    – J. M.'s eventual burnout ♦
    Mar 3, 2019 at 2:29

1 Answer 1

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Your series expansion for $\eta\to0$ is wrong:

Assuming[η > 0, Series[F[η], {η, 0, 30}]]

$0.000172184\frac{G_2}{G_{14}}η^{465/16} + \mathcal{O}(η)^{481/16}$

What you are seeing for $\eta<3$ is the finite number of terms you've included in your formula for $F$. If you include more terms, the "correct" behavior will extend further towards 0. But you cannot expect a series expansion around $\eta=+\infty$ to be accurate all the way down to $\eta=0$.

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  • $\begingroup$ Thank you for your explanation. My point is that the plot obtained (and correctly described by the series expansion that you have suggested) does not give the expected result of $F(\eta)$ in $\eta=0$, because I do know that it should be $F(0)\approx 3.23$ and not $0$. Of course this problem is due to the fact that the given function $F(\eta)$ has been defined using a product of serires expansions that are well-defined only for $\eta\to+\infty$. Is there a way to compute the domain of convergence of $F(\eta)$, that is the interval around $+\infty$ where the function has the correct trend? $\endgroup$
    – MariNala
    Mar 4, 2019 at 12:12
  • $\begingroup$ I don't know how to answer in general. en.wikipedia.org/wiki/Radius_of_convergence may be a good resource. $\endgroup$
    – Roman
    Mar 4, 2019 at 13:20

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