grandes-ecoles 2022 Q37

grandes-ecoles · France · centrale-maths2__psi Sequences and Series Power Series Expansion and Radius of Convergence

For $p \notin \mathbb{N}^*$, $p \neq 0$, let $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ be a non-zero power series solution of $(E_p) : x(y'' - y') + py = 0$, with coefficients satisfying $n(n+1)a_{n+1} = (n-p)a_n$ for all $n \in \mathbb{N}^*$. Show that there exists a natural integer $q > p$ such that, for all integer $n \geqslant q$, $$\left| a _ { n + 1 } \right| \geqslant \frac { \left| a _ { n } \right| } { 2 ( n + 1 ) }.$$

For $p \notin \mathbb{N}^*$, $p \neq 0$, let $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ be a non-zero power series solution of $(E_p) : x(y'' - y') + py = 0$, with coefficients satisfying $n(n+1)a_{n+1} = (n-p)a_n$ for all $n \in \mathbb{N}^*$. Show that there exists a natural integer $q > p$ such that, for all integer $n \geqslant q$,
$$\left| a _ { n + 1 } \right| \geqslant \frac { \left| a _ { n } \right| } { 2 ( n + 1 ) }.$$

Paper Questions

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