We denote by $\Sigma_{\infty}$ the set of sequences of real numbers $p = (p_{i})_{i \geqslant 1}$ such that $p_{i} \geqslant 0$ for all $i \geqslant 1$ and $\sum_{i=1}^{+\infty} p_{i} = 1$. We denote by $H_{\infty}$ the function on $\Sigma_{\infty}$ defined by $H_{\infty}(p) = \sum_{i=1}^{\infty} \varphi(p_{i})$ taking values in $\mathbb{R}_{+} \cup \{+\infty\}$, where $\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise.} \end{cases}$ (a) We consider $a \in ]0,1[$ and $p_{i} = a(1-a)^{i-1}$ for $i \geqslant 1$. Calculate $H_{\infty}(p)$ and study its variations as a function of $a$. (b) Show that there exists $p \in \Sigma_{\infty}$ such that $H_{\infty}(p) = +\infty$. (Hint: You may use without proof that the series with general term $n^{-1} \ln(n)^{-\beta}$ for $n \geqslant 2$ converges if and only if $\beta > 1$).
We denote by $\Sigma_{\infty}$ the set of sequences of real numbers $p = (p_{i})_{i \geqslant 1}$ such that $p_{i} \geqslant 0$ for all $i \geqslant 1$ and $\sum_{i=1}^{+\infty} p_{i} = 1$. We denote by $H_{\infty}$ the function on $\Sigma_{\infty}$ defined by $H_{\infty}(p) = \sum_{i=1}^{\infty} \varphi(p_{i})$ taking values in $\mathbb{R}_{+} \cup \{+\infty\}$, where $\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise.} \end{cases}$
(a) We consider $a \in ]0,1[$ and $p_{i} = a(1-a)^{i-1}$ for $i \geqslant 1$. Calculate $H_{\infty}(p)$ and study its variations as a function of $a$.
(b) Show that there exists $p \in \Sigma_{\infty}$ such that $H_{\infty}(p) = +\infty$. (Hint: You may use without proof that the series with general term $n^{-1} \ln(n)^{-\beta}$ for $n \geqslant 2$ converges if and only if $\beta > 1$).