grandes-ecoles 2020 Q27

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, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$. Determine the radius of convergence $R$ of the power series $\sum_{n \geqslant 1} a_n x^n$. 
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, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$. Determine the radius of convergence $R$ of the power series $\sum_{n \geqslant 1} a_n x^n$.
Paper Questions
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