Unexpected behavior of Series[ ]?

Questions
1

Hi, My troubles with using Series seem to never finish. Can anybody tell me why the following code produces an asymptotic series in powers of 1/x when nmax=50, but adds an oscillatory term involving Sin[x]*Cos[x] when nmax=60? Is this a correct and normal behaviour of Series[]? How can I verify if the correct expansion for real x involves the oscillatory terms?

F[x_]:=BesselJ[0,x]^2+BesselY[0,x]^2;
FullSimplify[Series[F[x],{x,Infinity,nmax},Assumptions->{x>0}],Assumptions->{x>0}]

Leslaw

2

I’m not seeing the Sin[x] * Cos[x] term when nmax = 60. I’m running Mathematica 12.2 on a Mac.

I do have a question though, if you want only real x’s should you add that to your assumptions using:

Assumptions -> {x > 0, Element[x, Reals]}
3

With Normal the oscillatory terms disappear:

F[x_] := BesselJ[0, x]^2 + BesselY[0, x]^2;
Series[F[x], {x, Infinity, 60}];
Simplify[% // Normal]
Limit[(F[x] - %)/(1/x^60), x -> Infinity]