Approximating distributions by finite number of moments Let $P$ be a distribution that is moment-determinate, i.e. it has finite moments of all orders and the infinite moments determine $P$ uniquely.
I was asking myself whether there are results on the approximation of $P$ from the first $N$ moments, $m_1, m_2, \dots, m_N$?
 A: You should probably refer to Casella and Berger, Exercises 2.34 and 2.35, as well as Section 2.6.1 after the exercises, where they discuss the uniqueness of moment sequences.  There are several references mentioned in the text, specifically that of Romano and Siegel (1986), Counterexamples in Probability and Statistics.  For your convenience, I have reproduced the content of these exercises below:
2.34
A distribution cannot be uniquely determined by a finite collection of moments, as this example from Romano and Siegel (1986) shows.  Let $X$ have the normal distribution, that is, $X$ has PDF $$f_X(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}, \quad -\infty < x < \infty.$$  Define a discrete random variable $Y$ by $$\Pr[Y = \sqrt{3}] = \Pr[Y = -\sqrt{3}] = \frac{1}{6}, \quad \Pr[Y = 0] = \frac{2}{3}.$$ Show that $$\mathrm{E}[X^r] =\mathrm{E}[Y^r], \quad \mathrm{for} \quad r = 1, 2, 3, 4, 5.$$  (Romano and Siegel point out that for any finite $n$ there exists a discrete and hence non-normal, random variable whose first $n$ moments are equal to those of $X$.)
2.35
Fill in the gaps in Example 2.3.10 below:
Example 2.3.10.  Consider the two PDFs given by $$\begin{align*} f_1(x) &= \frac{1}{\sqrt{2\pi} x} e^{-(\log x)^2/2}, \quad 0 \le x < \infty, \\ f_2(x) &= f_1(x) (1 + \sin(2\pi \log x)), \quad 0 \le x < \infty. \end{align*}$$  It can be shown that if $X_1 \sim f_1(x)$, then $\mathrm{E}[X_1^r] = e^{r^2/2}, \quad r = 0, 1, 2, \ldots,$ so $X_1$ has all of its moments.  Now suppose that $X_2 \sim f_2(x)$.  We have $$\mathrm{E}[X_2^r] = \int_{x=0}^\infty x^r f_1(x)(1 + \sin (2\pi \log x)) \, dx = \mathrm{E}[X_1^r] + \int_{x=0}^\infty x^r f_1(x) \sin (2\pi \log x) \, dx.$$  However, the transformation $y = \log x - r$ shows that this last integral is that of an odd function over $(-\infty, \infty)$ and hence is equal to zero for $r = 0, 1, 2, \ldots$.  Thus, even though $X_1$ and $X_2$ have distinct PDFs, they have the same moments for all $r$.  
