grandes-ecoles 2021 Q13

grandes-ecoles · France · centrale-maths1__official Sequences and series, recurrence and convergence Series convergence and power series analysis

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convenience $C_{0} = 1$. For every $x \in \left]-\frac{1}{4}, \frac{1}{4}\right[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Show that the function $f : \left]-\frac{1}{4}, \frac{1}{4}\right[ \rightarrow \mathbb{R},\; x \mapsto 2xF(x) - 1$ does not vanish.

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convenience $C_{0} = 1$. For every $x \in \left]-\frac{1}{4}, \frac{1}{4}\right[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.

Show that the function $f : \left]-\frac{1}{4}, \frac{1}{4}\right[ \rightarrow \mathbb{R},\; x \mapsto 2xF(x) - 1$ does not vanish.

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