I was asked to solve the following limit using the nth Taylor polynomial with remainder.

$$
\displaystyle{\lim\_{x \to 0}}\frac{\log{(1+x^2)}}{2x}.
$$

I couldn't find the remainder term because I couldn't generalize a formula for the nth derivative so I use the infinite series expansion

$$
-\sum\_{k=1}^{\infty}\frac{(-1)^k(x^{2})^k}{k}.
$$

What would be the way to do it with the remainder?