Exercise about linear operator

Question

For $X$ Banach, I have to show that if $T\in\mathfrak{L}(X)$ and $||T||_{\mathfrak{L}(X)}<1$ then exists $(I-T)^{-1}$ and $$ (I-T)^{-1}=\sum_{n=0}^\infty T^n. $$ For the existence of $(I-T)^{-1}$ I proved that $\ker(I-T)=\{0\}$. But for the second point of the proof I don't know how to proceed. I know that this is the geometric series and $||T||_{\mathfrak{L}(X)}<1$, but I'm not sure this is enough.

Answer 1

Look at this $$(I-T)\circ \sum_{j=0}^{n} T^j =I-T^{n+1} $$ and $$\left(\sum_{j=0}^{n} T^j \right)\circ (I-T)=I-T^{n+1} .$$