Look at the following expansion, which should be an expansion from for the coefficients $a_0, a_1, a_2, a_3$
$$\begin{align} z(\cos z -1) &= z \left( 1 - z^2/2!+ z^4/4! - z^6/6! + O\left(|z|^8\right) -1 \right)\\ &= z \left(-z^2/2! + O\left(|z|^4\right)\right)\\ &= -z^3/2 + O(|z|^5) \end{align}$$
Which results in
$$ a_0 = a_1 = a_2 = 0, a_3 = - \dfrac12$$
I'm having problem which understanding the $O$-term. Why is it present at all since, we're just interested in 4 terms anyway and why/how is it changed from $z^8$ to $z^4$ in one step.