<s>[INST] What is going wrong in this log expansion? I am getting a weird result here:
Let $p_1 = q_1 + \Delta$ and Let $p_2 = q_2 - \Delta$
I use the expansion $\log(1-x) = -x -x^2/2 -x^3/3 - ...$ in the third step. This expansion is valid for $x<1$ so there should be no problem with that.
\begin{align*}
    D(\vec{p}||\vec{q}) &= p_1\log{\frac{p_1}{q_1}} + p_2\log{\frac{p_2}{q_2}}\\
    &= p_1\log{\frac{q_1 + \Delta}{q_1}} + p_2\log{\frac{q_2 - \Delta}{q_2}}\\
    &= p_1(\frac{\Delta}{q_1} - \frac{\Delta^2}{q_1^2} + \frac{\Delta^3}{q_1^3} - ...) + p_2(-\frac{\Delta}{q_2} - \frac{\Delta^2}{q_2^2} - \frac{\Delta^3}{q_2^3} - ...)\\
    &= (q_1 + \Delta)(\frac{\Delta}{q_1} - \frac{\Delta^2}{q_1^2} + \frac{\Delta^3}{q_1^3} - ...) + (q_2 - \Delta)(-\frac{\Delta}{q_2} - \frac{\Delta^2}{q_2^2} - \frac{\Delta^3}{q_2^3} - ...)\\
    &=0
\end{align*}
The last equality follows from comparing terms of powers of $\Delta$ (am I not allowed to do that?).
 [/INST] There are at least two errors:
 $$p_1\log{\frac{q_1 + \Delta}{q_1}} = p_1\left(\frac{\Delta}{q_1} - \frac{\Delta^2}{\color{red}{2}q_1^2} - \frac{\Delta^3}{\color{red}{3}q_1^3} - \cdots\right)$$
Similar for the second term. 
And even if the low order terms cancel, the result is not zero. Using Maple I get (if I did not make mistakes)
$$D(\vec{p}||\vec{q}) = \frac{1}{2}\left(\frac{1}{q_1}+\frac{1}{q_2}\right) \Delta^2 + O(\Delta^3)$$
</s>