## 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.