grandes-ecoles 2020 Q34

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

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 $R$. It has been shown that $S(x) = W(x)$ for all $x \in ]-R,R[$. Does this result remain true 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 $R$. It has been shown that $S(x) = W(x)$ for all $x \in ]-R,R[$. Does this result remain true on $[-R, R]$?

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