Infinite sum of asymptotic expansions

Question

I have a question about an infinite sum of asymptotic expansions: Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$ with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq \dfrac{1}{k^3}$, $a_{2k}\leq \dfrac{1}{k^4}$, $\cdots$.

Does it follow that $\sum_{k=1}^{\infty}f_k(x)\sim \sum_{k=1}^{\infty}a_{0k}+\dfrac{\sum_{k=1}^{\infty}a_{1k}}{x}+\dfrac{\sum_{k=1}^{\infty}a_{2k}}{x^2}+\cdots$?

By the way, could you please suggest me materials to learn asymptotic expansions like this? Thanks

Answer 1

Of course it is possible, but much caution is required because we don't know about uniformity in $k$ for the estimates corresponding to the $\sim$ statements. Counterexamples are very easy to construct, where e.g. $f_k(x)$ doesn't start behaving "nicely" until $x > k$.