Hello. I was working on my complex analysis work, and I came across a problem that I am having significant difficulties with. The problem is basically to show that one formal power series is the inverse of another, given both of the power series. The ones I am working with now are actually exp(T) and ln(T). I know that these two series extend from the reals to the complex numbers uniquely, but I am pretty sure that the author wants it purely in the formal power series methods. Thanks!

Edit: I would much prefer hints, or at least a commentary on what I am doing than a correct answer. My method so far is:

I need to show that exp(log(1+T)) = 1+T. I am truncating log and exp at the N term, calling these exp<sub>N</sub>(T) and ln<sub>N</sub>(T). All that I need to show then is that exp<sub>N</sub>(ln<sub>N</sub>(1+T)) = 1+T+o(T<sup>N</sup>). Now I am going to try to expand (ln<sub>N</sub>(1+T))<sup>n</sup> and see if things cancel out when put into exp<sub>N</sub>(T).

Any pointers? Am I going down a dead end? Thanks.