grandes-ecoles 2019 Q15

grandes-ecoles · France · centrale-maths2__official Sequences and Series Recurrence Relations and Sequence Properties

For all $(x,n) \in \mathbb{R} \times \mathbb{N}$, we define $\pi_n(x) = \frac{\lfloor 2^n x \rfloor}{2^n}$ and $d_{n+1}(x) = 2^{n+1}(\pi_{n+1}(x) - \pi_n(x))$.
Establish $$\forall (x,j) \in \mathbb{R} \times \mathbb{N}^{\star}, \quad d_j(x) \in \{0,1\}.$$

For all $(x,n) \in \mathbb{R} \times \mathbb{N}$, we define $\pi_n(x) = \frac{\lfloor 2^n x \rfloor}{2^n}$ and $d_{n+1}(x) = 2^{n+1}(\pi_{n+1}(x) - \pi_n(x))$.

Establish
$$\forall (x,j) \in \mathbb{R} \times \mathbb{N}^{\star}, \quad d_j(x) \in \{0,1\}.$$

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