grandes-ecoles 2025 Q7

grandes-ecoles · France · mines-ponts-maths2__psi Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences

Let $k \in \mathbf { N } ^ { * }$. Show that $f \in \mathcal { C } ^ { k } \left( \mathbf { R } _ { + } , \mathbf { R } \right)$ and give an expression for $x \mapsto f ^ { ( k ) } ( x )$. Then express $f ^ { ( k ) } ( 0 )$ in terms of $b _ { k }$.
Here $f = \sum_{n\geq 0} f_n$ is the sum of a Dirichlet series with $f_n(x) = a_n e^{-\lambda_n x}$, and $b_k = \sum_{n=1}^{+\infty} \lambda_n^k a_n$.

Let $k \in \mathbf { N } ^ { * }$. Show that $f \in \mathcal { C } ^ { k } \left( \mathbf { R } _ { + } , \mathbf { R } \right)$ and give an expression for $x \mapsto f ^ { ( k ) } ( x )$. Then express $f ^ { ( k ) } ( 0 )$ in terms of $b _ { k }$.

Here $f = \sum_{n\geq 0} f_n$ is the sum of a Dirichlet series with $f_n(x) = a_n e^{-\lambda_n x}$, and $b_k = \sum_{n=1}^{+\infty} \lambda_n^k a_n$.

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