What is the domain of convergence in variable $a$ of the Taylor Series of this function?
$$h\left(a\right)\equiv-\int_{-\infty}^{\infty}p\left(x,a\right)ln\left(p\left(x,a\right)\right)dx$$ where
$$p\left(x,a\right)\equiv\frac{1}{2\sqrt{2\pi}}\left(e^{-(x+a)^{2}/2}+e^{-(x-a)^{2}/2}\right)$$
This is just the entropy of a bimodal distribution, expanding in the distance of the modes from the center. Here are the first several terms of the series, generated by Mathematica.
$$h\left(a\right)=\frac{1+ln\left(2\pi\right)}{2}+\frac{1}{2}a^{2}-\frac{1}{4}a^{4}+\frac{1}{6}a^{6}-\frac{5}{24}a^{8}+\frac{13}{30}a^{10}-\frac{227}{180}a^{12}+\frac{2957}{630}a^{14}-\frac{21425}{1008}a^{16}+\frac{642853}{5670}a^{18}+...$$