脑力劳动

2023-05-26 20:01 作者:虞廷采风人 0人读过 | 我要投稿

$S(x)%3D%5Csum_%7Bk%3D0%7D%5E%7B%E2%88%9E%7D%20%7B%20k%2Bm%2B1%20%5Cchoose%20m%2B1%20%7D%5Cfrac%7Bx%5Ek%20%7D%7Bk!%7D%20$

解：

$S(x)%3D%5Csum_%7Bk%3D0%7D%5E%7B%E2%88%9E%7D%20%7B%20k%2Bm%2B1%20%5Cchoose%20k%20%7D%5Cfrac%7Bx%5Ek%20%7D%7Bk!%7D%20%3D%5Csum_%7Bk%3D0%7D%5E%7B%E2%88%9E%7D%20%7B%20k%2Bm%20%5Cchoose%20k-1%20%7D%5Cfrac%7Bx%5Ek%20%7D%7Bk!%7D%20%2B%5Csum_%7Bk%3D0%7D%5E%7B%E2%88%9E%7D%20%7B%20k%2Bm%20%5Cchoose%20k%20%7D%5Cfrac%7Bx%5Ek%20%7D%7Bk!%7D%20$

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$%5Cimplies%20S'(x)-S(x)%3D%5Csum_%7Bk%3D0%7D%5E%7B%E2%88%9E%7D%20%7B%20k%2Bm%2B1%20%5Cchoose%20k%2B1%20%7D%5Cfrac%7Bx%5E%7Bk%20%7D%7D%7Bk!%7D$

$%7B%20k%2Bm%2B1%20%5Cchoose%20k%2B1%20%7D%3D%5Cfrac%7B(k%2Bm%2B1)!%7D%7Bm!(k%2B1)!%7D%20%3D%5Cfrac%7B(k%2Bm%2B1)!%7D%7Bk!(m%2B1)!%7D%5Cfrac%7Bm%2B1%7D%7Bk%2B1%7D%20%3D%5Cfrac%7Bm%2B1%7D%7Bk%2B1%7D%20%7B%20k%2Bm%2B1%20%5Cchoose%20m%2B1%20%7D$

$%5Cbegin%7Beqnarray*%7D%0A%5Cimplies%20%20x(S'(x)-S(x))%26%3D%26%5Csum_%7Bk%3D0%7D%5E%7B%E2%88%9E%7D%5Cfrac%7Bm%2B1%7D%7Bk%2B1%7D%20%7B%20k%2Bm%2B1%20%5Cchoose%20m%2B1%20%7D%5Cfrac%7Bx%5E%7Bk%2B1%20%7D%7D%7Bk!%7D%0A%5C%5C%26%3D%26(m%2B1)%5Csum_%7Bk%3D0%7D%5E%7B%E2%88%9E%7D%7B%20k%2Bm%2B1%20%5Cchoose%20m%2B1%20%7D%5Cfrac%7Bx%5E%7Bk%2B1%20%7D%7D%7B(k%2B1)!%7D%0A%5Cend%7Beqnarray*%7D$

$%5Cimplies%20%20%5Bx(S'(x)-S(x))%5D'%3D(m%2B1)%5Csum_%7Bk%3D0%7D%5E%7B%E2%88%9E%7D%7B%20k%2Bm%2B1%20%5Cchoose%20m%2B1%20%7D%5Cfrac%7Bx%5E%7Bk%20%7D%7D%7Bk!%7D%3D(m%2B1)S(x)$