由泰勒展开式知e^x= 1+x+x^2/2!+x^3/3!+...+x^n/n!+Rn(x)令x=1,得e=1+1+1/2!+1/3!+1/4!+......+1/n!+Rn(1)当n趋向于无穷时，Rn(1)为比x^n的高阶无穷小所以当n趋向于无穷时，e=1+1+1/2!+1/3!+1/4!+......+1/n!所以1/2!+1/3!+1/4!+......+1/n!=e-2(当n趋向于无穷时)
见下图
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