When do Taylor series approximations to expectations of (entire) functions converge? Take an expectation of the form $E(f(X))$ for some univariate random variable $X$ and an entire function $f(\cdot)$ (i.e., the interval of convergence is the whole real line)
I have a moment generating function for $X$ and hence can easily calculate integer moments.  Use a Taylor series around $\mu \equiv E(x)$ and then apply the expectation in terms of a series of central moments,
$$
E(f(x)) = E\left(f(\mu) + f'(\mu)(x - \mu) + f''(\mu)\frac{(x - \mu)^2}{2!} +\ldots\right)
$$
$$
 =f(\mu) + \sum_{n=2}^{\infty} \frac{f^{(n)}(\mu)}{n!}E\left[(x - \mu)^n\right]
$$
Truncate this series,
$$
E_N(f(x)) = 
f(\mu) + \sum_{n=2}^{N} \frac{f^{(n)}(\mu)}{n!}E\left[(x - \mu)^n\right]
$$

My question is: under what conditions on the random variable (and anything additional on $f(\cdot)$ as well) does the approximation of the expectation converge as I add terms (i.e. $\lim\limits_{N\to\infty}E_N(f(x)) = E(f(x))$).
Since it does not appear to converge for my case (a poisson random variable and $f(x) = x^{\alpha}$), are there any other tricks for finding approximate expectations with integer moments when these conditions fail?
 A: By your assumption that $f$ is real-analytic,
$$
y_n = f(\mu) + f'(\mu)(x - \mu) + f''(\mu)\frac{(x - \mu)^2}{2!} + \ldots + f^{(n)}(\mu)\frac{(x - \mu)^n}{n!}
$$
converges almost surely (in fact surely) to $f(x)$.
A standard condition under which a.s. convergence implies convergence of expectation, i.e.
$$
E[f(x)] = E [ \lim_{n \rightarrow \infty} y_n] = \lim_{n \rightarrow \infty} E [y_n],
$$
is that $|y_n| \leq y$ a.s. for some $y$ such that $E[y] < \infty$. (Dominated Convergence Theorem.)
This condition would hold if the power series converges absolutely a.s., i.e.
$$
y = \sum_{n \geq 0} |f^{(n)}(\mu)| \, \frac{|x - \mu|^n}{n!} < \infty  \;\; a.s.
$$
and 
$$
E[y] < \infty.
$$
Your example of a Poisson random variable and $f(x)=x^{\alpha}$, $\alpha \notin\mathbb{Z}_+$, would suggest that the above integrability of absolute limit criterion is the weakest possible, in general.
A: The approximation will converge if the function f(x) admits to power series expansion i.e. all derivatives exist. It also will be fully achieved if derivatives of a specific threshold and above are equal to zero. 
You can refer to Populis[3-4] and Stark and Woods [4]. 
