grandes-ecoles 2020 Q30

grandes-ecoles · France · centrale-maths2__psi Sequences and series, recurrence and convergence Series convergence and power series analysis

We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. Prove that $$\forall x \in ]-R, R[, \quad x(1 + S(x))S'(x) = S(x).$$ One may use the result from Question 26.

We define a sequence $(a_n)_{n \geqslant 1}$ by setting
$$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$
We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. Prove that
$$\forall x \in ]-R, R[, \quad x(1 + S(x))S'(x) = S(x).$$
One may use the result from Question 26.

Paper Questions

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