One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq n} \frac{1}{k}$, and failed. But should we necessarily fail?

More precisely, is it known that $H_n$

cannotbe written in terms of the elementary functions, say, the rational functions, $\exp(x)$ and $\ln x$? If so, how is such a theorem proved?

**Note**. When I started writing the question, I was going to ask if it is known that the harmonic function cannot be represented simply as a rational function? But this is easy to see, since $H_n$ grows like $\ln n+O(1)$, whereas no rational function grows logarithmically.

**Added note:** This earlier question asks a similar question for “elementary integration”. I guess I am asking if there is an analogous theory of “elementary summation”.