Global sections of $\mathcal{O}(k)$ over $\mathbb{P}^1$.

Question

I have the following example from Well's book on differential analysis on complex manifolds. I don't currently understand why they get the description for $\mathcal{O}(\Bbb P_1(\Bbb C), E^k)$ from the resulting power series.

For example with $k = 1$, this becomes $$w\sum_{n=0}^\infty b_nw^n = \sum_{n=0}^\infty b_nw^{n+1}$$ and this should reflect that there are no global sections in some sense?