用数学归纳法证明,1-x/1!+x(x-1)/2!+...+(-1)^nx(x-1)...(x-n

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  • 前面n=1时式子成立不写了

    假设n=k成立则1/x!+.(-1)^k x(x-1)(x-k+1)/k!=(-1)^k (x-1)(x-2)...(x-k)/k!成立

    则n=k+1时有1/x!+.(-1)^k x(x-1)(x-k+1)/k!+(-1)^(k+1) x(x-1)(x-k)/(k+1)!=(-1)^k (x-1)(x-2)...(x-k)/k!+(-1)^(k+1) x(x-1)(x-k)/(k+1)!=(-1)^(k+1) x(x-1)(x-k)/k!(k+1) - (-1)^(k+1) (x-1)(x-2)...(x-k)/k!=(-1)^(k+1) (x-1)(x-2)...(x-k)/k!*[x/(k+1)-1]=(-1)^(k+1) (x-1)(x-2)[x-(k+1)]/k!(k+1)=(-1)^(k+1) (x-1)(x-2)[x-(k+1)]/(k+1)!即当n=k+1时也成立;故式子得证