Show that the sequence $S_n := \sum_{k=0}^{n} \lambda_k A^k$ has a limit in $ L(X)$. I have a simple question, but I do not know what to do with it:
The question is as follows:
Let $X$ be a Banach space. Consider an integral operator $A \in L(X)$ and a function $\varphi(t) = \sum_{k=0}^{+\infty} \lambda_k t^k$ for $\lambda_k \in \mathbb{R}$. Where the series converges on the whole $\mathbb{R}$.
Prove that the sequence $S_n := \sum_{k=0}^{n} \lambda_k A^k$ has a limit $\varphi(A) \in L(X)$, when $n \to +\infty.$
The question looks simple, but I cannot understand it! It seems something is missing? Or we just need to show that $\|\sum_{k=0}^{n} \lambda_k A^k - \varphi(A) \| $ goes to zero in operator norm?
Can someone please let me know how can we show it?
Thanks!

The power series $\varphi(t)=\sum_{k=0}^{+\infty} \lambda_k t^k$ converges absolutely in each $t \in \mathbb R$.
Put $S_n:=\sum_{k=0}^{n} \lambda_k A^k$. For $n,m \in \mathbb N$ with $m>n$ we have
$||S_m-S_n||= ||\sum_{k=n+1}^{m} \lambda_k A^k|| \le \sum_{k=n+1}^{m} |\lambda_k| \cdot ||A||^k=|a_m-a_n|$, where $a_n:=\sum_{k=0}^{n} \lambda_k ||A||^k$
$(a_n)$ is convergent, hence  a Cauchy sequence.
This shows that $(S_n)$ is Cauchy sequence in $L(X)$.
Since $X$ is a Banach space, we have that $L(X)$ is a Banach space.
Conclusion ?
