grandes-ecoles 2022 Q6

grandes-ecoles · France · mines-ponts-maths1__mp Sequences and Series Power Series Expansion and Radius of Convergence

Let $z \in D$. We agree that $p _ { n , 0 } = 0$ for all $n \in \mathbf { N }$. By examining the summability of the family $\left( \left( p _ { n , N + 1 } - p _ { n , N } \right) z ^ { n } \right) _ { ( n , N ) \in \mathbf { N } ^ { 2 } }$, prove that
$$P ( z ) = \sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n }$$
Deduce the radius of convergence of the power series $\sum _ { n } p _ { n } x ^ { n }$.

Let $z \in D$. We agree that $p _ { n , 0 } = 0$ for all $n \in \mathbf { N }$. By examining the summability of the family $\left( \left( p _ { n , N + 1 } - p _ { n , N } \right) z ^ { n } \right) _ { ( n , N ) \in \mathbf { N } ^ { 2 } }$, prove that

$$P ( z ) = \sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n }$$

Deduce the radius of convergence of the power series $\sum _ { n } p _ { n } x ^ { n }$.

Paper Questions

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