Let $k \in \mathbf { N } ^ { * }$. Show that $f \in \mathcal { C } ^ { k } \left( \mathbf { R } _ { + } , \mathbf { R } \right)$ and give an expression for $x \mapsto f ^ { ( k ) } ( x )$. Then express $f ^ { ( k ) } ( 0 )$ in terms of $b _ { k }$.
Here $f = \sum_{n\geq 0} f_n$ is the sum of a Dirichlet series with $f_n(x) = a_n e^{-\lambda_n x}$, and $b_k = \sum_{n=1}^{+\infty} \lambda_n^k a_n$.

Show that if $f ( x ) = 0$ for all $x \in \mathbf { R } _ { + }$ then $a _ { n } = 0$ for all $n \in \mathbf { N }$.
Here $f = \sum_{n\geq 0} a_n e^{-\lambda_n x}$ is the sum of a Dirichlet series.

Show that
$$\left\| y _ { N } - y \right\| _ { \infty , \mathbf { R } _ { + } } \leq \frac { M } { 2 ^ { N } }$$
and deduce that $y _ { N }$ converges uniformly to $y$ on $\mathbf { R } _ { + }$. Then propose an interval $J \subset \mathbf { R } _ { + }$ where the bound on $\left\| y _ { N } - y \right\| _ { \infty , J }$ would be sharper.
Here $y _ { N } ( x ) = \sum _ { n = 0 } ^ { N } a _ { n } \mathrm { e } ^ { - \lambda _ { n } x }$ and $y ( x ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \mathrm { e } ^ { - \lambda _ { n } x }$ with $\left| a_n \right| \leq \frac{M}{2^n}$.