One of the new features in Mathematica 12.1 is the function
Asymptotic. Here’s a quick example of using it.
Here’s an asymptotic series for the log of the gamma function I wrote about here.
If we ask Mathematica
Asymptotic[LogGamma[z], z -> Infinity]
we get simply the first term:
But we can set the argument
SeriesTermGoal to tell it we’d like more terms. For example
Asymptotic[LogGamma[z], z -> Infinity, SeriesTermGoal -> 4]
This doesn’t contain a term 1/z4, but it doesn’t need to: there is no such term in the asymptotic expansion, so it is giving us the terms up to order 4, it’s just that the coefficient of the 1/z4 term is zero.
If we ask for terms up to order 5
Asymptotic[LogGamma[z], z -> Infinity, SeriesTermGoal -> 5]
we do get a term 1/z5, but notice there is no 4th order term.
A note on output forms
The Mathematica output displayed above was created by using
to save images as SVG files. The alt text for the images was created using