Interchanging the Order of Integrals, Infinite Series, Derivatives, and Limits

You can view the document here:

https://drive.google.com/file/d/1Xrwpct54z88Q_uTmWARd9MYKdVs4k0L9/view?usp=drive_link

Often in math youtube channels when they’re solving problems involving calculus, there comes a time when they want to switch the order of, say an integral and derivative, or some other pair of operators, but they don’t explain why this is justified. Often, it is not justified, unless the problem is carefully crafted to be. This document goes over every strategy I could find to justify switching the order of any pair of the four standard calculus operators, including when they’re paired with themselves. It includes the conditions, definitions when necessary, and examples.