>Would topics such as: Strings, formal languages, finite state machines, Turing machines, primitive recursive and recursive functions, and recursive unsolvability, be considered as mathematics or just exclusively Computer Science?
There is a substantial intersection between the set of things we label *mathematics* and the set of things we call *computer science*. All the topics you mention fall very firmly within that intersection.
>Is it worth studying the theory of algorithms as an undergrad math major?
Are you interested in the theory of algorithms? If you are, then it's definitely worth it.
At a math department I worked at 10 years ago, I used to teach a class which covered some of the topics you mentioned above. I can tell you it was a fun class to teach and the students were generally interested in the subject.