My question isn't about a specific mathematical problem. It's about the comparison method in general. I've been studying some real analysis and I specially struggle with the convergence test problems, and have realised that most of them can be solved easily with comparison test. I totally understand the method and how it works, but I have this questions:
- Is there any "typical form" of series $\sum a_n$ that I can recognize and say: "This one is about comparison test" ? (For example, for the series $\sum (a_n)^n$ I intuitively go for the root test, so I want to find a pattern like that but for comparison test).
- Is there any "rule" or method I can follow to easily find the right sequences to compare with? (I've seen answers in my textbook that use comparison test with sequences that seem totally impossible to guess, unless I'm missing something about how to find that "good" sequences).
I'm not pretty sure if this will make any sense, but I struggle with this method of convergence testing and maybe there's something I'm ignoring that makes it easier. Any help will be very appreciated.