A function $f(n)$, defined for positive integer $n$, has an asymptotic expansion for large $n$ of the following form:

$$f(n) \sim \sum_{k=0}^{\infty} a_{k} \frac{1}{n^{2 k}}, \quad n \rightarrow \infty$$

What precisely does this mean?

Show that the integral

$$I(n)=\int_{0}^{2 \pi} \frac{\cos n t}{1+t^{2}} d t$$

has an asymptotic expansion of the form $(*)$. [The Riemann-Lebesgue lemma may be used without proof.] Evaluate the coefficients $a_{0}, a_{1}$ and $a_{2}$.