grandes-ecoles 2022 Q20

grandes-ecoles · France · mines-ponts-maths1__psi Sequences and series, recurrence and convergence Series convergence and power series analysis

For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ such that $\sum_{k=1}^{N} k a_k = n$. If this set is finite, we denote by $p_{n,N}$ its cardinality. We denote by $p_n$ the final value of $(p_{n,N})_{N \geq 1}$.
Let $z \in D$. By examining the difference $\sum_{n=0}^{+\infty} p_n z^n - \sum_{n=0}^{+\infty} p_{n,N} z^n$, prove that $$P(z) = \sum_{n=0}^{+\infty} p_n z^n$$

For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ such that $\sum_{k=1}^{N} k a_k = n$. If this set is finite, we denote by $p_{n,N}$ its cardinality. We denote by $p_n$ the final value of $(p_{n,N})_{N \geq 1}$.

Let $z \in D$. By examining the difference $\sum_{n=0}^{+\infty} p_n z^n - \sum_{n=0}^{+\infty} p_{n,N} z^n$, prove that
$$P(z) = \sum_{n=0}^{+\infty} p_n z^n$$

Paper Questions

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