求证1!+22!+33!+…+n*n!=(n+1)!-1

来源：学生作业帮编辑：大师作文网作业帮分类：数学作业时间：2024/11/12 20:13:00

求证1!+2*2!+3*3!+…+n*n!=(n+1)!-1

证：
1!+2×2!+3×3!+...+n×n!
=1!+(3-1)×2!+(4-1)×3!+...+(n+1-1)×n!
=1!+3!+4!+...+(n+1)!-2!-3!-...-n!
=1!+(n+1)!-2!
=(n+1)!+1-2
=(n+1)!-1
等式成立.

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