for those who have calculus by apostol vol.1, i refer to page 288. i am looking at the first example where he proves that tanx = x + (1/3)x3 + o(x3). he showed that 1/cosx = 1 + (1/2)x2 + o(x2) and therefore tanx = sinx / cosx = (x - (1/6)x3 + o(x4))(1 + (1/2)x2 + o(x2)) and that should equal x + (1/3)x3 + o(x3). i multiplied it out and got x + (1/3)x3 + o(x3) - (1/12)x5 - o(x5) +o(x4) + o(x6) + o(x4)o(x2). my question is where did the rest of the o's go and how come you are only left with o(x3)? why not o(x4)?