设 $\sum_{n=1}^{\infty} u_n = s$ 收敛。考察 $\sum_{n=1}^{\infty} (u_n - u_{n+1} + 2u_{n+2})$。将其拆分为三个级数：$\sum u_n - \sum u_{n+1} + 2\sum u_{n+2}$。因原级数收敛，其各项移位后仍收敛，且 $\sum_{n=1}^{\infty} u_{n+1} = s - u_1$，$\sum_{n=1}^{\infty} u_{n+2} = s - u_1 - u_2$。代入得：$s - (s - u_1) + 2(s - u_1 - u_2) = s - s + u_1 + 2s - 2u_1 - 2u_2 = 2s - u_1 - 2u_2$，对应选项 C。选项 A、B、D 的系数或常数项均不匹配该推导结果。