Question #41298

Let $f_k(x) = \sum_{n=1}^{\infty} n^k x^{n-1}$ for $|x| < 1$ and $k \in \{0, 1, 2, 3\}$. Define the determinants: $\Delta_1(x) = \begin{vmatrix} f_1(x) & f_2(x) & f_3(x) \\ f_1(1/2) & f_2(1/2) & f_3(1/2) \\ f_1(-1/2) & f_2(-1/2) & f_3(-1/2) \end{vmatrix}$ $\Delta_2(x) = \begin{vmatrix} f_0(x) & f_1(x) & f_2(x) \\ f_0(1/2) & f_1(1/2) & f_2(1/2) \\ f_0(-1/2) & f_1(-1/2) & f_2(-1/2) \end{vmatrix}$ Find the value of $\lim_{x \to 1/2} \frac{\Delta_1(x)}{\Delta_2(x)}$.