For an operator $T \in B(X)$, its spectral radius is $$r(T) = \lim_{n\rightarrow \infty} \|T^n\|^{1/n}$$ and the Neumann series is $$\sum_{n=0}^{\infty}T^n.$$
If $r(T)>1$, I can show that series is not absolutely convergent $\left(\sum_{n=0}^{\infty}\|T^n\| \rightarrow \infty\right)$ but I'm not sure how to show it isn't convergent, i.e. $\sum_{n=0}^{\infty}T^n \notin B(X)$. It would certainly suffice to show that $$\left\lbrace \sum_{n=0}^{k} T^n \right \rbrace_{k=1}^\infty$$ is not Cauchy or not bounded but I cant think of any appropriate lower bounds. Is there a general strategy for proving a series in a normed space is not convergent?