There are many discontinuous Point derivations on the Banach algebra $(\mathbb{C^{(n)}[0,1], \|\|_n})$

This is an Exercise 6.2.55 in Garth Dales , Introduction to Banach algebra

Show that there are many discontinuous Point derivations on the Banach algebra $ (\mathbb{C^{(n)}[0,1], \|\|_n})$ where $ $ \|f\|_n=\sum_{k=0}^{n}\frac{1}{k!}|f^{(k)}|$ $ for all $ f\in \mathbb{C^{(n)}[0,1]} $

so if you give reasonable hints, I will be very happy. Thanks

Find all the $c \ge 0$ for which $\sum_{n=1}^{+ \infty }a _{n}$ is absolutely convergent

We consider: $ $ a_{1}=c-1$ $ $ $ a_{n+1}= \frac{-n}{n+c \cdot \sqrt[n]{ln(n^{9876}+17)}}\cdot a_{n}, n\ge 1$ $ $ $ c\ge0$ $ I want to use Rabbe Test, because then in a simple way it comes out that the series is convergent for every $ c\ge0$ . However I have two doubts: 1) Raabe Test is for $ a_{n}>0$ , but if I do $ r_{n}=n(|\frac{a_{n}}{a_{n+1}}|-1)$ I knew that I must removing minus at $ n$ and then I can leave the module. Hovewer I’m not sure if it’s allowed. 2) If Raabe Test is a good way to do this task I knew only when my series is convergent, but I don’t knew when is absolutely convergent so the more I do not know if Raabe Test is a good idea.