grandes-ecoles 2017 QIID

grandes-ecoles · France · centrale-maths1__pc Sequences and Series Power Series Expansion and Radius of Convergence

We set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts. The sequence $\left( \frac { B _ { n } } { n ! } \right) _ { n \in \mathbb { N } }$ is bounded by 1.
Deduce a lower bound for the radius of convergence $R$ of the power series $\sum _ { n \geqslant 0 } \frac { B _ { n } } { n ! } z ^ { n }$.

We set for every integer $n \geqslant 0$,
$$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$
where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts. The sequence $\left( \frac { B _ { n } } { n ! } \right) _ { n \in \mathbb { N } }$ is bounded by 1.

Deduce a lower bound for the radius of convergence $R$ of the power series $\sum _ { n \geqslant 0 } \frac { B _ { n } } { n ! } z ^ { n }$.

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