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求极限（x+x2+…+x^n-n）/(x-1)
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求极限（x+x2+…+x^n-n）/(x-1)
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答案和解析
①罗必塔法则：
lim(x->1) (x+x^2+…+x^n-n）/(x-1)
=lim(x->1) (1+2x+3x^2+…+nx^(n-1)）/1
=[1+2+...+n]
=n(n+1)/2
②初等变形：
lim(x->1) [(x-1)+(x^2-1)+…+(x^n-1)] /(x-1)
=lim(x->1) [(x-1)/(x-1)+(x^2-1)/(x-1)+…+(x^n-1)/(x-1)]
=lim(x->1) 1+(x+1)+(x^2+x+1)+…+(x^(n-1)+x^(n-2)+...+x^2+x+1)
=1+2+3+...+n
=n(n+1)/2
lim(x->1) (x+x^2+…+x^n-n）/(x-1)
=lim(x->1) (1+2x+3x^2+…+nx^(n-1)）/1
=[1+2+...+n]
=n(n+1)/2
②初等变形：
lim(x->1) [(x-1)+(x^2-1)+…+(x^n-1)] /(x-1)
=lim(x->1) [(x-1)/(x-1)+(x^2-1)/(x-1)+…+(x^n-1)/(x-1)]
=lim(x->1) 1+(x+1)+(x^2+x+1)+…+(x^(n-1)+x^(n-2)+...+x^2+x+1)
=1+2+3+...+n
=n(n+1)/2
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