What is the Taylor expansion of $\arctan$ for a power series

Question

My function is $x = A/t + B + C \cdot t + D \cdot t^2 + O(t^3)$.

What is the $\arctan(x)$ series expansion for small $t$?

I know the $\arctan$ expansion for small and large values, but here the first term is large, the second term is constant, and the other terms are small.

So I do not know what is the asymptotic expansion for this. Thanks for any help.

Answer 1

$$x=\frac At+B+Ct+Dt^2+O(t^3)=\frac At\left(1+\frac BAt+\frac CAt^2+\frac DAt^3+O(t^4)\right)$$ and

$$\frac1x=\frac tA\left(1-\frac BAt-\left(\frac{B^2}{A^2}+\frac CA\right)t^2-\cdots\right)$$ which you can plug in $\pi/2-\arctan1/x$.