Brilliant Series!

Consider $f_n(x) = 3(1-x)^n + 5\dfrac {d} {dx} (1-x)^n$

$\begin{aligned} f_n(x) & = 3 \left[ \begin{pmatrix} n \\ 0 \end{pmatrix} - \begin{pmatrix} n \\ 1 \end{pmatrix} x + \begin{pmatrix} n \\ 2 \end{pmatrix} x^2-...(-1)^n \begin{pmatrix} n \\ n \end{pmatrix} x^n \right] \\ & \quad + 5 \dfrac {d}{dx} \left[ \begin{pmatrix} n \\ 0 \end{pmatrix} - \begin{pmatrix} n \\ 1 \end{pmatrix} x + \begin{pmatrix} n \\ 2 \end{pmatrix} x^2-...(-1)^n \begin{pmatrix} n \\ n \end{pmatrix} x^n \right] \\ & = 3 \left[ \begin{pmatrix} n \\ 0 \end{pmatrix} - \begin{pmatrix} n \\ 1 \end{pmatrix} x + \begin{pmatrix} n \\ 2 \end{pmatrix} x^2-...(-1)^n \begin{pmatrix} n \\ n \end{pmatrix} x^n \right] \\ & \quad + 5 \left[ -\begin{pmatrix} n \\ 1 \end{pmatrix} + 2\begin{pmatrix} n \\ 2 \end{pmatrix} x - 3\begin{pmatrix} n \\ 3 \end{pmatrix} x^2-...(-1)^n n \begin{pmatrix} n \\ n \end{pmatrix} x^{n-1} \right] \\ & = 3 \begin{pmatrix} n \\ 0 \end{pmatrix} - (3x+5) \begin{pmatrix} n \\ 1 \end{pmatrix} + (3x^2+10x) \begin{pmatrix} n \\ 2 \end{pmatrix} \\ & \quad -(3x^3+15x^2) \begin{pmatrix} n \\ 3 \end{pmatrix} +...(-1)^n(3x^n+5nx^{n-1}) \begin{pmatrix} n \\ n \end{pmatrix} \end{aligned}$

We note that:

$\begin{aligned} f_n(1) & = 3 \begin{pmatrix} n \\ 0 \end{pmatrix} -8 \begin{pmatrix} n \\ 1 \end{pmatrix} + 13 \begin{pmatrix} n \\ 2 \end{pmatrix} -18 \begin{pmatrix} n \\ 3 \end{pmatrix} +...\pm(3+5n) \begin{pmatrix} n \\ n \end{pmatrix} \\ & = = 3(1-1)^n + 5\dfrac {d} {dx} (1-1)^n = \boxed{0} \end{aligned}$