EDB — 1TW

Exercises

1. [1TW]Let \(f(x)=∑_{n=0}^∞ a_ n x^ n\) with radius of convergence \(𝜌{\gt}0\), and let \(f(0)=f'(0)=\ldots =f^{(n)}(0)=0\); show that the function \(g(x)=f(x) / x^ n\) is extendable to \(x=0\); show that (the extension of) \(g\) coincides with an appropriate power series \(g(x)=∑_{n=0}^∞ b_ n x^ n\). What can be said about the radius of convergence of \(g\)?