不会推的式子

· · 个人记录

\sum_{x=1}^n m(m-1)^{x-1}\sum_{i=0}^{n} (-1)^i \binom{x}{i} \binom{n-ik-1}{x-1}\\ =\sum_{x=1}^n m(m-1)^{x-1}\sum_{i=0}^{n} (-1)^i \left[\binom{x-1}{i} + \binom{x-1}{i-1}\right] \binom{n-ik-1}{x-1}. \begin{aligned}& \sum_{x=1}^n m(m-1)^{x-1}\sum_{i=0}^{n} (-1)^i \binom{x-1}{i} \binom{n-ik-1}{x-1}\\ = & \sum_{x=1}^n m(m-1)^{x-1}\sum_{i=0}^{n} (-1)^i \binom{n-ik-1-i}{x-1-i}\binom{n-ik-1}{i}\\ = & m \sum_{i=0}^{n} (-1)^i \binom{n-ik-1}{i} \sum_{x=1}^n (m-1)^{x-1} \binom{n-ik-1-i}{x-1-i}\\ = & m \sum_{i=0}^{n} (-1)^i (m-1)^i \binom{n-ik-1}{i} \sum_{x=1}^n (m-1)^{x-1-i} \binom{n-ik-1-i}{x-1-i}\\ = & m \sum_{i=0}^{n} m^{n-i(k+1)-1} (1-m)^i \binom{n-ik-1}{i}.\end{aligned} & \sum_{x=1}^n m(m-1)^{x-1}\sum_{i=1}^{n} (-1)^i \binom{x-1}{i-1} \binom{n-ik-1}{x-1}\\ = & \sum_{x=1}^n m(m-1)^{x-1}\sum_{i=1}^{n} (-1)^i \binom{n-ik-i}{x-i} \binom{n-ik-1}{i-1}\\ = & m \sum_{i=1}^{n} (-1)^i \binom{n-ik-1}{i-1} \sum_{x=1}^n (m-1)^{x-1} \binom{n-ik-i}{x-i}\\ = & m \sum_{i=1}^{n} (-1)^i (m-1)^{i-1} \binom{n-ik-1}{i-1} \sum_{x=1}^n (m-1)^{x-i} \binom{n-ik-i}{x-i}\\ = & -m \sum_{i=1}^{n} m^{n-i(k+1)} (1-m)^{i-1} \binom{n-ik-1}{i-1}. \end{aligned} \sum^{+\infty} = \frac{+\infty}{2}