TITLE: Why would you take the logarithmic derivative of a generating function?
QUESTION [14 upvotes]: Today, my climbing expedition scaled Mt. Sloane to request the Oracle's Extensive Insight into Sequences. The monks there had never heard of our plight, so they inscribed our query in mystical runes on a scrip of paper and took it into a room we were not permitted to enter. The Superseeker, as they called it, eventually responded with a fresh scroll, bearing (among other, more familiar, symbols) six imposing letters: LGDEGF.
"Logarithmic Derivative Exponential Generating Function," the monks muttered in unison as I unravelled the scroll, nodding and tittering amongst themselves. But what is such a thing? They were quick to recite that it is a function $f$ such that
$$\exp\biggl(\int f(x) \,dx\biggr) = \sum_n a_n \frac{x^n}{n!}$$
for my sequence $a_n$, and that the information in the scroll pertained to this $f$, but they refused to answer any further questions.
My expedition crew was well-versed in the basic science of generating functions, ordinary power series and exponential. But why might taking the logarithmic derivative of either generating function give interesting or exciting information? Where do they occur in the wild? Most importantly, where in the literature can we learn about them?

REPLY [4 votes]: One thing to add on to Michael's excellent answer. 
A random variable $X$ (with all cumulants) is normal iff $\kappa_n(X)=0$ for all $n\ge 3$. Furthermore, a sequence of random variables $X_k$ (with all cumulants) converges in distribution to a normal distribution iff $\kappa_n(X_k)\to 0$ for all $n\ge 3$. 
This is often in practice much easier to show than to show all moments converge to respective moments of a normal distribution.