I am aware that there are some techniques which can be used to show that some function does not have an antiderivative expressible using elementary functions, such as Liouville's theorem. (More broadly, this falls into the area of differential Galois theory and differential algebra. Such topics were discussed also on this site, for example, here or here.)
Are there some analogous results for finite sums? More precisely, suppose we have a sum $$s(n)=\sum_{k=1}^n f(k)$$ where $f$ is a given elementary function. Are there some methods which can be used to show that $s(n)$ is not equal to an elementary function (restricted to $\mathbb N$)?
To give a specific example, let us take $f(n)=\frac1n$. Are there some methods which can be used to show that we cannot express the $n$-the harmonic number $H_n=\sum_{k=1}^n\frac1k$ as $H_n=s(n)$ for an elementary function $s(x)$? (And is such result for harmonic numbers even known?)
EDIT: Very shortly after posting this question I found this: Do harmonic numbers have a “closed-form” expression?
Which definitely answers the second part about harmonic numbers. (Gerry Myerson's answer mentions also others sums, not only $H_n$. So it can definitely be considered as an answer for this question, too. We will see whether somebody will post some additional interesting information as an answer to this question.)