I'm reading a chapter in "Economic and Financial Decisions under Risk" which quickly covers the foundations for things like "constant relative risk aversion." At one point, this involves using a first-order approximation of one side of an equation and a second order approximation for the other side.
In the following, $u$ is a utility function over wealth $w$, $z$ is a "risky lottery" (something which will increase or decrease wealth) with mean zero and variance $\sigma^2$, and $e$ is some constant change in wealth which is the "certainty equivalent" of $z$. We want: $$ \begin{aligned} \mathbb{E}\left[u(w+z)\right]=u(w+e) \end{aligned} $$ The authors use a second-order approximation for the LHS: $$ \begin{aligned} \mathbb{E}\left[u(w+z)\right] &\approx\mathbb{E}\left[u(w)+zu^\prime(w)+z^2u^{\prime\prime}(w)/2\right]\\ &=u(w)+u^\prime(w)\mathbb{E}[z]+u^{\prime\prime}(w)\mathbb{E}[z^2]/2\\ &=u(w)+u^{\prime\prime}(w)\sigma^2/2 \end{aligned} $$ And then a first-order approximation for the RHS: $$ \begin{aligned} u(w+e)&\approx u(w)+u^\prime(w)e \end{aligned} $$ To then get the approximation $e\approx \sigma^2u^{\prime\prime}(w)/2u^\prime(w)$.
How should I think about the validity of these approximations? Is there some deeper reason why these particular orders were used (beyond "the resulting solution is tractable")? When would I expect them to break down, and to what degree? (In this case the authors have their own discussion on the matter, but I'd like to know what to be looking out for in general.)
I've tried out using a second-order approximation for the RHS above, which leads to some square roots when solving for $e$, and so I am tempted to replace those square roots with the Taylor series for $\sqrt{1+x}$. When does a "nested approximation" like that make sense?
Throwing Pade approximants into the mix, when is it "reasonable" to use a Pade approximant for one part of a problem and a Taylor series for another?
I'd also be happy with some pointers to good introductory books or tutorials on this.