The only explanations I've seen of the divergence of the arctangent taylor series outside $R=1$ have to do with arctangent having an infinity at $i$. I find this interesting because it almost seems like it forces "$i$" to be a very real thing if we want to explain whats going on. However, I've also noticed that many problems which have solutions involving complex numbers have counterparts which don't involve them; complex numbers just seems to make things easier. So, is there another way of doing this?
Here is one example of an explanation I've seen involving complex numbers: Power series representation of arctangent: fails to converge everywhere