I want to prove the existence of $n \in \mathbb{Z}^+$ and $a_0,a_1,\dots,a_n \in \mathbb{R}$ such that
$$\bigg| \left( \sum_{k=0}^n \dfrac{a_k}{x^k} \right) - \exp \left( \dfrac{\sin(e^x)}{\sqrt{x}} \right) \bigg| \leq 10^{-6}, \quad \forall x \in [1,\infty).$$
I was thinking on trying Darboux's Riemann integrable definition, or some kind of root/ratio test. I am sorry not giving some work, but I am really without a clue on how to solve this one. Any ideas? Thank you.
With a suitable change of variable, we are just looking for a polynomial approximation $p(x)$ of $$ f(x)=\exp\left(\sqrt{x}\sin e^{\frac{1}{x}}\right) $$ over the interval $I=(0,1]$, such that $\|p-f\|_{\infty}\leq 10^{-6}$.
Despite the wild behaviour of $f(x)$ on I:
$f(x)$ is still a continuous function over $I$, hence Weierstrass' approximation theorem is enough to ensure the existence of $p(x)$.