Suppose I have a theory that gives me a series expansion for some function $f(x;\lambda)$, assuming large $\lambda$, in the form $$f(x;\lambda)=\sum_{n=0}^\infty \frac{c_n(x)}{\lambda^n}.$$
I claimed that $f(x;\lambda)=c_0+c_1/\lambda+O(\lambda^{-2}),$ but someone told me I was confusing "the $O(\cdot)$ asymptotic Landau-notation with a perturbative expansion" and that "convergence only holds weakly".
I am not sure what this means.
My question is exactly what terminology and notation I should use in this case.