Definition of Absolute Risk Aversion In its Wikipedia article, absolute risk aversion is defined as 
$ARA = -\frac{u''(c)}{u'(c)}$. However, I have alternatively seen absolute risk aversion
defined as half the decrease in consumption that an investor is willing to accept to avoid a gamble $\varepsilon$, where $E[\varepsilon] = 0$, $E[\varepsilon^2] = 1$, and $\varepsilon$
is independent of consumption:
$$
U(C - ARA/2) \equiv E[U(C + \varepsilon) \mid C].
$$
Using the latter definition, how can I show that $ARA = -U''(C) / U'(C)$?
Set $y \equiv c+\varepsilon$. So $y$ represents changed consumption around and "close" to a given level $c$. Take a 2nd-order Taylor expansion of the function $E[u(y)\mid c]$  around $c$, which is treated as fixed since we condition on it :

$$E[u(y)\mid c] \approx E[u(c)\mid c] + E[u'(c)(y-c)\mid c] + E[\frac12u''(c)(y-c)^2\mid c]$$

But $y-c = \varepsilon$
so

$$E[u(y)\mid c] \approx E[u(c)\mid c] + E[u'(c)\varepsilon\mid c] + E[\frac12u''(c)\varepsilon^2\mid c]$$

Due to the conditioning, and the independence of $\varepsilon$ from $c$ the expected value distributes and applies only to $\varepsilon$:

$$E[u(y)] \approx u(c) + u'(c)E[\varepsilon] + \frac12u''(c)E[\varepsilon^2]$$

Since we assume 

$$E[\varepsilon] = 0 \Rightarrow {\rm Var}(\varepsilon)=E[\varepsilon^2] =1$$
we obtain

$$E[u(c+\varepsilon)] \approx u(c) + \frac12u''(c)  \tag{1}$$

Now consider $u(c-\frac 12 ARA)$, with $ARA \equiv -\frac{u''(c)}{u'(c)}$, and take a 1st-order Taylor expansion in this case, again around $c$:

$$u\left(c-\frac 12 ARA\right) = u(c) + u'(c)\cdot (c- \frac 12 ARA - c) = u(c) - \frac 12 u'(c)\cdot ARA $$

Using the definition of $ARA$ to replace it we obtain

$$u\left(c-\frac 12 ARA\right) \approx u(c) - \frac 12 u'(c)\cdot\left (-\frac{u''(c)}{u'(c)}\right)$$

$$\Rightarrow u\left(c-\frac 12 ARA\right) \approx u(c) + \frac 12 \cdot u''(c) \tag{2}$$

The right-hand sides of equations $(1)$ and $(2)$ are equal therefore, approximately, so  are their left-hand sides or,

$$E[u(c+\varepsilon) \mid c] \approx u\left(c-\frac 12 ARA\right)$$

which is valid as long as $ARA$ is defined the way it is. QED.  

If the variance of the gamble is not unity but $\sigma^2 \neq 1$, then the more general equation is

$$E[u(c+\varepsilon) \mid c] \approx u\left(c-\frac {\sigma^2}2 ARA\right)$$