grandes-ecoles 2022 Q9

grandes-ecoles · France · centrale-maths2__mp Sequences and Series Proof of Inequalities Involving Series or Sequence Terms

We fix $f \in \mathcal{C}^{K}([0,1])$ and denote by $P$ the polynomial determined in question Q7. Deduce the inequality $$\left\|f^{(k)} - P^{(k)}\right\|_{\infty} \leqslant \left\|f^{(k+1)} - P^{(k+1)}\right\|_{\infty}$$ for all $k \in \llbracket 0, K-1 \rrbracket$.

We fix $f \in \mathcal{C}^{K}([0,1])$ and denote by $P$ the polynomial determined in question Q7. Deduce the inequality
$$\left\|f^{(k)} - P^{(k)}\right\|_{\infty} \leqslant \left\|f^{(k+1)} - P^{(k+1)}\right\|_{\infty}$$
for all $k \in \llbracket 0, K-1 \rrbracket$.

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23 Q24 Q25 Q26 Q27 Q28 Q29 Q30 Q31 Q32 Q33 Q34