Question

请教一道数学题目：求下列函数的极限。

Answer 1

cosx = 1 - x^2 / 2 + o(x^2)
(cosx)^n = 1 - nx^2 / 2 + o(x^2)
那么 lim (cosx + .... + (cosx)^n - n) / (cosx - 1)
= (-1/2 + -2/2 + ... + -n/2) / (-1/2) = n(n+1)/2

Answer 2

x^n-y^n=(x-y)[x^(n-1)+x^(n-2)y+x^(n-3)y^2+...+x^2y(n-3)+xy^(n-2)+y^(n-1)]

cosx+(cosx)^2+.....+(cosx)^n-n=cosx-1+(cosx)^2-1+....+(cosx)^n-1

=(cosx-1)(1+cosx+1+....(cosx)^n-1+(cosx)^n-2+.....cosx+1)

所以原式=lim(x->0)(1+cosx+1+....(cosx)^n-1+(cosx)^n-2+.....cosx+1)
=1+2+3+....n=n(1+n)/2

Answer 3