grandes-ecoles 2018 Q17

grandes-ecoles · France · x-ens-maths2__mp Sequences and Series Evaluation of a Finite or Infinite Sum

We denote by $n$ the integer part of $\frac{N}{2}$. For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$.
Give the explicit formula for $R_{N}$, in terms of the polynomials $P_{j}$.

We denote by $n$ the integer part of $\frac{N}{2}$. For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$.

Give the explicit formula for $R_{N}$, in terms of the polynomials $P_{j}$.

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