高数之麦克劳林

tanx = sinx/cosx

=(x-x^3/6+x^5/120+o(x^5))/(1-[x^2/2-x^4/24+o(x^4)])

=(x-x^3/6+x^5/120+o(x^5))*{(1+ [x^2/2+x^4/24+o(x^4)]+[x^2/2+x^4/24+o(x^4)]^2+o(x^4)}

=(x-x^3/6+x^5/120+o(x^5))*{(1+ [x^2/2+x^4/24+o(x^4)]+x^4/4+o(x^4))}

=(x-x^3/6+x^5/120+o(x^5))*{(1+ x^2/2+7x^4/24+o(x^4)}

= x + x^3/3 + 2/15*x^5 + o(x^5)

① tanx = x + o(x^2)

② tanx = x + x^3/3 + o(x^4)

③ tanx = x + x^3/3 + 2/15*x^5 + o(x^6) （∵奇函数）