Show that the series of functions $\sum u_k$ where for all $k \in \mathbb{N}^*$, the function $u_k$ is defined on $[0, +\infty[$ by $u_k : x \mapsto (1 + kx)^{-k}(1/2)^k$ is normally convergent on $[0, +\infty[$.
Show that the series of functions $\sum u_k$ where for all $k \in \mathbb{N}^*$, the function $u_k$ is defined on $[0, +\infty[$ by $u_k : x \mapsto (1 + kx)^{-k}(1/2)^k$ is normally convergent on $[0, +\infty[$.