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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

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    $\begingroup$ Doesn't the ratio test predict divergence outside $R=1$? $\endgroup$ – Matthew Leingang Mar 12 '18 at 22:18
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    $\begingroup$ Please cite an explicit reference. "The only explanations I've seen" isn't useful. $\endgroup$ – Rob Arthan Mar 12 '18 at 22:18
  • $\begingroup$ @RobArthan ; Thank you, I've added an explanation I've seen $\endgroup$ – Richard Wilde Mar 12 '18 at 22:27
  • $\begingroup$ I certainly can't think of one, and the observation that 1/(x^2 + 1) doesn't have a series that converges on all of R makes it unlikely that you'll find one. This is a bit like the fact that every real polynomial of degree 3 or more has a quadratic factor. It's true, but try proving it without complex numbers. $\endgroup$ – Mike Housky Jun 28 '18 at 21:39

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