consider two functions $f,g:(0,1)\rightarrow\mathbb{R}$ which have the same asymptotic power series for $t\downarrow 0$, i.e. $f(t)\sim\sum_k a_k t^{k}$ and $g(t)\sim\sum_k a_k t^{k}$ as $t\downarrow 0$ for some coefficients $a_k.$
One way to write this relation would be $f(t)=g(t)+o(t^k),$ $\forall k\geq 0$.
I am wondering if I could just abbreviate this as $f(t)\sim g(t)$ as $t\downarrow 0$ ? I am not sure if both notations would be equivalent or whether some information is lost through the second notation?
Best wishes
The notation $f(t) \sim g(t)$ (which means that $\lim_{t \to 0^+} f(t)/g(t) = 1$) would only tell you that the first nonzero terms in the asymptotic expansions for $f(t)$ and $g(t)$ are equal (presuming you already know that the expansions are of the same form).
As far as I know there is not a common notation which means that two functions share the same asymptotic expansion. Your expression using $o(t^k)$ seems fine to me.