x的n次方-1有没有公式?
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x的n次方-1有没有公式?
l i m x的n次方-1
----------=
x→1 x-1
l i m x的n次方-1
----------=
x→1 x-1
![x的n次方-1有没有公式?](/uploads/image/z/6117516-36-6.jpg?t=x%E7%9A%84n%E6%AC%A1%E6%96%B9-1%E6%9C%89%E6%B2%A1%E6%9C%89%E5%85%AC%E5%BC%8F%3F)
x^n=(x-1+1)^n
=(x-1)^n+n(x-1)^(n-1)+.+n(x-1)^1+1
则(x^n-1)/(x-1)=(x-1)^(n-1)+n(x-1)^(n-2)+.n(n-1)(x-1)/2+n
则lim(x->1):(x^n-1)/(x-1) =n
2.
x^n-1=(x-1)(x^(n-1)+x^(n-2)+.+1)
lim(x-->1):(x^n-1)/(x-1)=lim(x^(n-1)+x^(n-2)+.+1)=n
=(x-1)^n+n(x-1)^(n-1)+.+n(x-1)^1+1
则(x^n-1)/(x-1)=(x-1)^(n-1)+n(x-1)^(n-2)+.n(n-1)(x-1)/2+n
则lim(x->1):(x^n-1)/(x-1) =n
2.
x^n-1=(x-1)(x^(n-1)+x^(n-2)+.+1)
lim(x-->1):(x^n-1)/(x-1)=lim(x^(n-1)+x^(n-2)+.+1)=n