grandes-ecoles 2017 QIIIA

grandes-ecoles · France · centrale-maths1__pc Sequences and Series Properties and Manipulation of Power Series or Formal Series

We define the sequence of polynomials $\left( H _ { k } \right) _ { k \in \mathbb { N } }$ in $\mathbb { R } [ X ]$ by $H _ { 0 } ( X ) = 1$ and, for all $k \in \mathbb { N } ^ { * }$, $$H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$$
Show that the family $( H _ { 0 } , \ldots , H _ { n } )$ is a basis of the space $\mathbb { R } _ { n } [ X ]$.

We define the sequence of polynomials $\left( H _ { k } \right) _ { k \in \mathbb { N } }$ in $\mathbb { R } [ X ]$ by $H _ { 0 } ( X ) = 1$ and, for all $k \in \mathbb { N } ^ { * }$,
$$H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$$

Show that the family $( H _ { 0 } , \ldots , H _ { n } )$ is a basis of the space $\mathbb { R } _ { n } [ X ]$.

Paper Questions

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