grandes-ecoles 2021 Q10

grandes-ecoles · France · centrale-maths1__official Sequences and Series Power Series Expansion and Radius of Convergence

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convenience $C_{0} = 1$.
Show that, for every natural integer $n$, $C_{n} \leqslant 2^{2n}$. What can we deduce for the radius of convergence of the power series $\sum C_{k} x^{k}$?

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convenience $C_{0} = 1$.

Show that, for every natural integer $n$, $C_{n} \leqslant 2^{2n}$. What can we deduce for the radius of convergence of the power series $\sum C_{k} x^{k}$?

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