Asymptotic expansion of $\cos x$

I want to determine whether or not $$\cos x$$ has an asymptotic expansion of the form $$\sum_{k=0}^{\infty} \frac{a_k}{x^k}$$ as $$x \to \infty$$ in $$\mathbb{R^+}$$.

This means we are taking the asymptotic series $$\psi_k(x) = x^{-k}$$, but I don't really know how to go about starting this.

• $\cos\left(\frac{1}{x}\right)$ is not a holomorphic function in a neighbourhood of zero: it has an essential singularity there. Commented May 18, 2017 at 18:59
• @JackD'Aurizio But my function is $\cos(x)$, why can we look at $\cos\left(\frac{1}{x}\right)$? Oh, is it that we can look at $\cos(1/x)$ as x tends to 0 as that's the same as $\cos(x)$ as x tends to infinity? Commented May 18, 2017 at 19:01
• @user112495 $\lim_{x\to \infty}\cos(x)$ does NOT exist. Commented May 18, 2017 at 19:01
• Apply the change of variable $x\to\frac{1}{x}$. Assuming that $\cos(x)$ has such expansion, it follows that $\cos\frac{1}{x}$ has a Taylor series at the origin. Commented May 18, 2017 at 19:02
• @JackD'Aurizio very instructive argument! Commented May 18, 2017 at 19:02

$$\lim_{n\to+\infty}\cos (2n\pi)=1$$
$$\lim_{n\to+\infty}\cos (\frac {\pi}{2}+n\pi)=0$$
thus $\lim_{x\to+\infty}\cos (x)$ doesn't exist and so does $\lim_{x\to+\infty}x^n\cos (x)$.