Let be (E, $\Vert .\Vert$) a $\Bbb K$-banach space , $(c_k)_k$ a sequence in E , $\beta$ the convergence radius of $\sum_{k=0}^\infty c_k \lambda^k$ . In my script is $\beta$ defined by Cauchy-Hadamard formula.
Prove: $\beta$=sup{|$\lambda$|: $\lambda$ $\in$ $\Bbb K$ , $(\sum_{k=0}^n c_k \lambda^k)_n$ bounded in E }=:s .
My thoughts:
To show 1) s$\le$ $\beta$ 2) $\beta$ $\le$ s .
To 1) : Let $a_n:=\sum_{k=0}^n c_k \lambda^k$, M:={$\lambda$ $\in$$\Bbb K$ :$(a_n)_n$ bounded} , $S_n$:=$\sum_{k=0}^n \Vert c_k \lambda^k \Vert $. If I can show |$\lambda$|$\le$$\beta$ ,$\forall$ $\lambda$ $\in$M,then I'm done. Assume: $\exists$ $\lambda$ $\in$M with |$\lambda$|>$\beta$. Because $\beta$ is convergence radius , for C>0 there is N $\in \Bbb N$ with $S_n>C ,\forall n\ge N$ . To show: $\lambda \notin M$ . That means t.s: $(a_n)_n$ is unbounded. T.s: For $C>0, \exists n \in \Bbb N$ with $\Vert a_n \Vert>C$. (*)
Do you know, how can you show (*) ? Or, do you have another suggestion, how can be solved the task ?