https://en.wikipedia.org/wiki/Big_O_notation

↑のランダウの記号の説明に、

e^x
= 1 + x + x^2/2 + O(x^3) as x -> 0
= 1 + x + O(x^2) as x -> 0

などと書かれていますが、

e^x = 1 + x + x^2/2 + O(x^2) as x -> 0
= 1 + x + O(x) as x -> 0

が正しいですよね。