(08/13/2022, 08:48 AM)tommy1729 Wrote: I got inspired by this nested root topic at MSE
https://math.stackexchange.com/questions...tx2-0?rq=1
regards
tommy1729
Fuck, that's a good thread. I really don't think it's as hard as they're making it though. For the simply reason, you can find constants:
\[
f_n(x-q_n) = g_n(x)\\
\]
Which centers the solution for \(\Re(x) > 0\). Then we can check that, for \(x_n \approx 0\), that:
\[
\sum_{n=0}^\infty |g_n(x_n)| < \infty\\
\]
Because:
\[
\sum_{n=0}^\infty |g'_n(x)|< \infty
\]
This tells us that \(g_n(x_n)\) converges; which casts a net of values about \(0\). It won't be a neighborhood. But instead it'll mean that: The function \(g(x)\) is real analytic, and probably holomorphic for \(\Re(x) > 0\). I think the fact no one in this thread is mentioning that, is something crucial to the problem. I could go into detail on why this converges, but it would require me explaining about 3 years worth of infinite compositions.