grandes-ecoles 2022 Q24

grandes-ecoles · France · centrale-maths2__pc Taylor series Determine radius or interval of convergence

Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$ and $r \in ] 0 , R [$. Show that there exists $C \in \mathbb { R }$ such that $$\forall k \in \mathbb { N } , \quad \left| c _ { k } \right| \leqslant \frac { C } { r ^ { k } }.$$

Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$ and $r \in ] 0 , R [$. Show that there exists $C \in \mathbb { R }$ such that
$$\forall k \in \mathbb { N } , \quad \left| c _ { k } \right| \leqslant \frac { C } { r ^ { k } }.$$

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