grandes-ecoles 2020 Q29

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 the function $S$ is defined and continuous on $[-R, R]$. 
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 the function $S$ is defined and continuous on $[-R, R]$.
Paper Questions
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