When it comes to proving any derivative, the first thing that comes to mind is induction, which is definitely a step. You have to slowly prove the existence of derivatives from the first and second order, that is, the left and right derivatives of the ratio limit exist and are equal in the open interval, and then use induction. You can't find the analytical expression of this problem, and you can't find such a function, so you should use σ (- 1) n/n x to represent the sum function. Remember, the point of derivation can only be an interior point, or it is the same to prove differentiability! In fact, the term of this series is an exponential function, so you have to prove that the exponential function continues at (0, ∞).