Let $\alpha$ be a real strictly greater than 1. The Riemann series $\sum_{n \geqslant 1} \frac{1}{n^{\alpha}}$ converges to a real that we will denote $\ell$. We denote by $\left(S_{n}\right)_{n \in \mathbb{N}}$ the sequence defined by $S_{0}=0$ and $\forall n \geqslant 1, S_{n}=\sum_{k=1}^{n} \frac{1}{k^{\alpha}}$. a) Show that $\forall n \geqslant 1, \frac{1}{\alpha-1} \frac{1}{(n+1)^{\alpha-1}} \leqslant \ell-S_{n} \leqslant \frac{1}{\alpha-1} \frac{1}{n^{\alpha-1}}$. b) Deduce that $\left(S_{n}\right)_{n \in \mathbb{N}}$ belongs to $E^{c}$ and give its convergence rate.
Let $\alpha$ be a real strictly greater than 1. The Riemann series $\sum_{n \geqslant 1} \frac{1}{n^{\alpha}}$ converges to a real that we will denote $\ell$. We denote by $\left(S_{n}\right)_{n \in \mathbb{N}}$ the sequence defined by $S_{0}=0$ and $\forall n \geqslant 1, S_{n}=\sum_{k=1}^{n} \frac{1}{k^{\alpha}}$.
a) Show that $\forall n \geqslant 1, \frac{1}{\alpha-1} \frac{1}{(n+1)^{\alpha-1}} \leqslant \ell-S_{n} \leqslant \frac{1}{\alpha-1} \frac{1}{n^{\alpha-1}}$.
b) Deduce that $\left(S_{n}\right)_{n \in \mathbb{N}}$ belongs to $E^{c}$ and give its convergence rate.