I found in the literature the following expansion in power series.
Given the regular function $A(k^2)$: $$A(k^2)=A(k^2=0)+ \frac{dA}{dk}(k^2=0)k^2+A_c(k^2)k^2$$ with $A_c(k^2)$ that goes at least linearly to zero when $k \longrightarrow 0$
Notice that $A_c(k^2)$ is not evaluated in $k=0$.
I was used to consider the Taylor expansion as: $$A(k^2)=A(k^2=0)+ \frac{dA}{dk}(k^2=0)k^2+ o(k^4)$$
If the expression are equivalent how to see that.