Consider the field of Hahn series (see e.g. https://en.wikipedia.org/wiki/Hahn_series )$$K[[T^{\mathbb{Q}}]]=\left\{\sum_{q\in\Bbb{Q}} a_q T^q: a_q\in K,\text{supp}(a_q)\text{ is well ordered} \right\}$$ ($\text{supp}(a_q)=\{q\in\Bbb{Q}:a_q\ne0\}$)with usual addition and multiplication: $$\sum_{q\in\Bbb{Q}}a_q T^q+\sum_{q\in\Bbb{Q}} b_q T^q=\sum_{q\in\Bbb{Q}} (a_q+b_q) T^n$$ and $$\left(\sum_{q\in\Bbb{Q}} a_q T^q \right)\left(\sum_{q\in\Bbb{Q}} a_q T^q\right)=\sum_{q\in\Bbb{Q}} \sum_{q_1+q_2=q} a_{q_1}b_{q_2} T^q$$ and with the norm $$\left\|\sum_{q} a_q T^q \right\|=\rho^{\min\{q\;:\;a_q\ne 0\}}$$ where $0<\rho<1$.
I conclude that the series $$\ln(T)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}(T-1)^n$$ does not belong to $K[[T^{\mathbb{Q}}]]$, because, while the sequence $$x_n=\sum_{m=1}^n \frac{(-1)^{m+1}}{m}(T-1)^m$$ is a sequence in $K[[T^{\Bbb{Q}}]]$, but it is not Cauchy sequence, therefore from this strategy, we can't conclude that $\ln$ belongs to $K[[T^{\Bbb{Q}}]]$ or not.
And I stack here.
Please help me to figure out if $\ln$ belongs to $K[[T^{\Bbb{Q}]]$ or not.
Thanks.