Taylor expansion of a function of two variables, one of which is an exponent

Question

Which is the Taylor series of the second order, centered at the origin, of the function:

$$ f(x,y)=(1+x^3)^y $$

Answer 1

You might write your function as $ f(x,y) = \exp(y \log(1+x^3))$. Now $\log(1+x^3) = O(x^3)$, so $$f(x,y) = \exp(O(x^3 y)) = 1 + O(x^3 y)$$ Thus the Taylor expansion to second order is just $1$.