Solution
= Solution
For the <arctangent>, put $h=x-1$. Its <derivative> is $(d/dh)\arctan(1+h)=1/(2+2h+h^2)=\frac12-\frac h2+\frac{h^2}4+O(h^4)$. Integrating and using $\arctan1=\pi/4$ gives
$$
\boxed{\arctan x=\frac\pi4+\frac h2-\frac{h^2}4+\frac{h^3}{12}+O(h^5).}
$$
The constant counts as the first nonzero term. The quartic coefficient vanishes; the next nonzero term is $-h^5/40$.