How would you use series to evaluate the lim(x-0) of (x-arctanx)/x^3? I'm getting stuck, and the answer says it's 1/3 but I don't know how they got there.

Question

How would you use series to evaluate the lim(x->0) of (x-arctanx)/x^3? I'm getting stuck, and the answer says it's 1/3 but I don't know how they got there.

Steve · Answer

arctan x = x - x^3/3 + x^5/5 - ... so, x-arctan(x) = x^3/3 - x^5/5 + x^7/7 - ... divide by x^3 and you get 1/3 - x^2/5 + x^5/7 - ... as x->0, that -> 1/3