<s>[INST] What is the Taylor's expansion of $f(x)=\exp(\frac{1}{2}c^2x^2)$ with a constant $c$? What is the Taylor's expansion of $f(x)=\exp(\frac{1}{2}c^2x^2)$ with a constant $c$.
Note that $f'(x)=\exp(\frac{1}{2}c^2x^2)c^2x$ and $f''(x)=\exp(\frac{1}{2}c^2x^2)(c^2x)^2+\exp(\frac{1}{2}c^2x^2)c^2$. Then $f'(0)=0$ and $f''(0)=c^2$.
Is that
$$
f(x)=f(0)+f'(0)x+\frac{1}{2}f''(0)x^2+o(x^2)=1+\frac{1}{2}c^2(\frac{1}{2}c^2x^2)^2?
$$
 [/INST] Just use Taylor expansion as if it were $e^X$ where $X = \frac{1}{2}c^2x^2$ hence:
$$e^X = \sum_{k = 0}^{+\infty} \frac{X^k}{k!}$$
$$e^{\frac{1}{2}x^2x^2} = \sum_{k = 0}^{+\infty} \frac{\left(\frac{1}{2}c^2x^2\right)^k}{k!} =\sum_{k = 0}^{+\infty}\frac{1}{2^k k!} (cx)^{2k}$$
The first terms are given by
$$1+\frac{c^2 x^2}{2}+\frac{c^4 x^4}{8}+O\left(x^5\right)$$
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