# 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. – murray Mar 2 at 21:30
• Can you maybe give a reference to the book/paper where this divergent series came from? – J. M. will be back soon Mar 3 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? – MariNala Mar 4 at 12:12