2nd-order taylor series of function of vector

Question

$$ f(y) = \frac{(y_{2}-1)^{2}}{(y_{1}+1)^{3}} $$

What would the 2nd-order Taylor formula be for $f(0+z)$ in terms of $z_{1}$ and $z_{2}$?

$y$ is a $2 \times 1$ vector

Answer 1

If $\vert x\vert < 1$ then $\displaystyle\sum_{k=0}^{\infty}(-1)^kx^k = \frac{1}{x+1}$.

Also we have $\frac{d^2}{dx^2}\frac{1}{(x+1)}=\frac{2}{(x+1)^3}$.

So $\frac{1}{(y_1+1)^3} = \frac{1}{2}\displaystyle\sum_{k=2}^{\infty}(-1)^k k(k-1)y_1^{k-2}$

and finally

$f(y) = \frac{1}{2}(y_2-1)^2\displaystyle\sum_{k=2}^{\infty}(-1)^k k(k-1)y_1^{k-2}$