grandes-ecoles 2022 Q18

grandes-ecoles · France · centrale-maths2__mp Sequences and series, recurrence and convergence Series convergence and power series analysis

Let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. Justify the existence of a strictly increasing sequence of natural integers $(\phi(j))_{j \in \mathbb{N}}$ satisfying $$\forall j \in \mathbb{N}, \quad \sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}.$$

Let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. Justify the existence of a strictly increasing sequence of natural integers $(\phi(j))_{j \in \mathbb{N}}$ satisfying
$$\forall j \in \mathbb{N}, \quad \sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}.$$

Paper Questions

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