How does Mathematica take a Series expansion at Infinity?

Question

I have a program that requires a series expansion at infinity and at a finite horizon, and I have posted two questions about simplifying these expressions or obtaining the series expansion recently. However, I now realize that I am a little confused about how this expansion even works in Mathematica, since I do not understand some of the test results I have, and I have gone through a few questions such as 1, 2 but I am still a little confused.

Consider the following examples I am obtaining (running Mathematica 11.3 on Windows 10)

In:  Series[Sin[x],{x,x0,3}]
Out: Sin[x0]+Cos[x0] (x-x0)-1/2 Sin[x0] (x-x0)^2-1/6 Cos[x0] (x-x0)^3+O[x-x0]^4

In:  Series[f[x],{x,x0,3}]
Out: f[x0]+(f^\[Prime])[x0] (x-x0)+1/2 (f^\[Prime]\[Prime])[x0] (x-x0)^2+1/6 (f^(3))[x0] (x-x0)^3+O[x-x0]^4

These examples I understand. However, at infinity:

In:  Series[Sin[x],{x,\[Infinity],3}]
Out: Sin[x+O[1/x]^5]

Since sin(x) is bounded, everywhere, I do not understand what this output means.

In:  Series[f[x],{x,\[Infinity],3}]
Out: f[x]

At infinity, I do not get symbolic series expansions as well. I understand that the regular Taylor series expansion definition around infinity is not meaningful, however, if you look at the second Mathematica notebook in this link (titled Amplification factors of the superradiant scattering of a neutral bosonic wave of generic spin off a Kerr BH, obtained by solving the Teukolsky equations), there is a section of the program that performs a symbolic series expansion at infinity, and I am able to run it and get results as well. So I'm confused about the conditions under which series expansions at infinity return meaningful results, since I think I do not know how to either give the right input, or understand the output from Mathematica. I'm quite new to this.

For reference, the actual code I need help with is here where the Series command just runs indefinitely with no results, or errors.

Answer 1

Let us consider an example, where the power expansion at infinity exists, e.g.

Normal[Series[(x - 1)/(3 x + 1), {x, x0, 3}]]

$\frac{36 (x-\text{x0})^3}{(3 \text{x0}+1)^4}-\frac{12 (x-\text{x0})^2}{(3 \text{x0}+1)^3}+\frac{4 (x-\text{x0})}{(3 \text{x0}+1)^2}+\frac{\text{x0}-1}{3 \text{x0}+1}$

The above makes no sense at infinity because no arithmetical operation with Infinity is defined. The limit of the above at infinity, i.e. 1/3, is too poor. I think Mathematica follows usual way of calculus by the change of a variable

Normal[Series[(x - 1)/(3 x + 1) /. x -> 1/y, {y, 0, 3}]] /. y -> 1/x

$-\frac{4}{81 x^3}+\frac{4}{27 x^2}-\frac{4}{9 x}+\frac{1}{3}$

The above is identical with the result of

Normal[Series[(x - 1)/(3 x + 1), {x, Infinity, 3}]]