I audited my Intro to Computer science course for four weeks before I was able to get officially enrolled into the course. I am now a computer science major.
If you didn’t get into your course, but you’re still dead set on taking it, here’s what you can do:
If you are unfamiliar with the professor, ask around. Their previous students might have a good idea about how lenient they are about the enrollment cap, or how easy/difficult they are to persuade based on the professor’s personality or their personal experience with the professor.
Some professors are very strict about the cap and won’t even allow auditing, in which case, you will have to accept this and wait for the next semester. Otherwise, we can continue with our scheme.
Keep showing up. Of course, show up for all the classes during Shopping Week, but also keep showing up under the pretense of auditing the class. Communicate with your professor about wanting to audit the class, should you not be able to get in. If applicable, attend all lab sessions and recitation sessions that you can make during that first week, and after that, attending just one lab session and/or recitation like any other enrolled student is enough. Make sure the professor knows you’re there. If they have a sign-in sheet, that will be easy.
However, although this is a good way to impress your professor with your dedication to the subject, this is not a guarantee. I am recommending this under the impression that whether or not you get enrolled in the class officially, you are happy to take it, even if it is just to audit it. At least next time you take the course, this time as an officially enrolled student, you’ll have an easier time with the material.
Let people know your situation. Stay in contact with the professor, the TAs, and your dean.