Binomial sums and series in code

Two ideas of this lesson are easy to test in code: the sum of $r\,{}^{n}C_{r}$, and the coefficient of $x^k$ in $(1 + x)^{\alpha}$ for an index $\alpha$ that is not a whole number. A falling product replaces the factorial.

Your tasks.

1. In weighted, return ${}^{n}C_{1} + 2\,{}^{n}C_{2} + \dots + n\,{}^{n}C_{n}$, using a loop.
2. In closed, return the closed form $n\,2^{n - 1}$.
3. In series_coefficient, return the coefficient of $x^k$ in $(1 + x)^{\alpha}$ as a Fraction: multiply $\dfrac{\alpha - i}{i + 1}$ for $i$ from 0 to $k - 1$.