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 $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq \dfrac{1}{k^3}$, $a_{2k}\leq \dfrac{1}{k^4}$, $\cdots$.

Is this possible $\sum_{k=1}^{\infty}f_k(x)\sim \sum_{k=1}^{\infty}a_{0k}+\dfrac{\sum_{k=1}^{\infty}a_{1k}}{x}+\dfrac{\sum_{k=1}^{\infty}a_{2k}}{x^2}+\cdots$?

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

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