I am self-studying “An Introduction to Multivariable Mathematics” by Leon Simon. While studying the section on power and Taylor series, I have encountered a question on the “consequence” of the following theorem:
If $f(x)=\sum_{n=0}^\infty a_n x^n$ has radius of convergence $\rho>0$, and if $\lvert \alpha \rvert<\rho$, then we can write$$f(x)=\sum_{m=0}^\infty b_m (x-\alpha)^m,$$where $\lvert x-\alpha \rvert < \rho - \lvert \alpha \rvert$ and $b_m=\sum_{n=m}^\infty {n\choose m} a_n {\alpha}^{n-m}$.
The note states that part of its conclusion implies that each $b_m$ converges. However, I am not sure how it must be the case. For instance, could it be the case that each $b_m$ diverges but $\sum_{m=0}^\infty b_m (x-\alpha)^m$ converges?