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 $\left| a_n \right| \leq \frac{M}{2^n}$.
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 $\left| a_n \right| \leq \frac{M}{2^n}$.