Do harmonic numbers have a “closed-form” expression?

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 Hn=kn1k, and failed. But should we necessarily fail?

More precisely, is it known that Hn cannot be written in terms of the elementary functions, say, the rational functions, exp(x) and lnx? 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 Hn grows like lnn+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”.

Answer

There is a theory of elementary summation; the phrase generally used is “summation in finite terms.” An important reference is Michael Karr, Summation in finite terms, Journal of the Association for Computing Machinery 28 (1981) 305-350, DOI: 10.1145/322248.322255. Quoting,

This paper describes techniques which greatly broaden the scope of what is meant by ‘finite terms’…these methods will show that the following sums have no formula as a rational function of n:
ni=11i,ni=11i2,ni=12ii,ni=1i!

Undoubtedly the particular problem of Hn goes back well before 1981. The references in Karr’s paper may be of some help here.

Attribution
Source : Link , Question Author : Srivatsan , Answer Author : Martin Sleziak

Leave a Comment