# Divergent series not correctly plotted

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 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]]

• 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. Mar 2, 2019 at 21:30
• Can you maybe give a reference to the book/paper where this divergent series came from? Mar 3, 2019 at 2:29

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 $$\eta=0$$.
• 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? Mar 4, 2019 at 12:12