grandes-ecoles 2021 Q30

grandes-ecoles · France · centrale-maths2__pc Sequences and Series Properties and Manipulation of Power Series or Formal Series
We consider a power series $\sum_{n \geqslant 0} \alpha_n z^n$, with radius of convergence $R \neq 0$, $\alpha_0 = 1$, and sum $S$.
Show that $\frac{1}{S}$ is expandable as a power series in a neighbourhood of 0.
We consider a power series $\sum_{n \geqslant 0} \alpha_n z^n$, with radius of convergence $R \neq 0$, $\alpha_0 = 1$, and sum $S$.

Show that $\frac{1}{S}$ is expandable as a power series in a neighbourhood of 0.