2011-07-20 05:03:34 8 Comments

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”.

### Related Questions

#### Sponsored Content

#### 2 Answered Questions

### [SOLVED] Is there a closed form for the alternating series of inverse harmonic numbers?

**2017-09-14 09:36:46****Dr. Wolfgang Hintze****322**View**9**Score**2**Answer- Tags: sequences-and-series harmonic-numbers

#### 1 Answered Questions

### [SOLVED] Infinite series with harmonic numbers related to elliptic integrals

**2018-02-02 07:47:05****Nemo****105**View**2**Score**1**Answer- Tags: sequences-and-series special-functions generating-functions harmonic-numbers elliptic-integrals

#### 2 Answered Questions

### [SOLVED] Symmetry relation for finite sums of generalized harmonic numbers

**2018-01-04 09:39:10****Dr. Wolfgang Hintze****70**View**1**Score**2**Answer- Tags: summation harmonic-numbers

#### 2 Answered Questions

### [SOLVED] Closed form expressions for harmonic sums

**2017-04-19 11:56:51****Przemo****305**View**2**Score**2**Answer- Tags: sequences-and-series discrete-mathematics harmonic-numbers

#### 6 Answered Questions

### [SOLVED] Simple proof of showing the Harmonic number $H_n = \Theta (\log n)$

**2013-02-17 18:01:30****mayank****20098**View**15**Score**6**Answer- Tags: sequences-and-series asymptotics harmonic-numbers

#### 0 Answered Questions

### The Harmonic Logarithm and its relation to the Prime Number Theorem

**2017-08-28 15:42:16****James Arathoon****96**View**1**Score**0**Answer- Tags: sequences-and-series elementary-number-theory prime-numbers harmonic-numbers

#### 2 Answered Questions

### [SOLVED] Are there some techniques which can be used to show that a sum "does not have a closed form"?

**2016-04-15 09:30:35****Martin Sleziak****229**View**16**Score**2**Answer- Tags: summation closed-form elementary-functions

#### 0 Answered Questions

### A family of generating functions related to generalized harmonic numbers and power functions.

**2016-09-08 15:49:42****Przemo****52**View**1**Score**0**Answer- Tags: summation closed-form

#### 1 Answered Questions

### [SOLVED] Harmonic Identity Evidence

**2015-07-30 06:27:02****Dom****84**View**1**Score**1**Answer- Tags: number-theory harmonic-numbers

#### 1 Answered Questions

### [SOLVED] Prove that primitives of $\frac{x^3}{{\rm e}^x - 1}$ have no closed form in terms of elementary functions

**2014-12-14 08:34:52****omegadot****153**View**5**Score**1**Answer- Tags: integration closed-form

## 5 comments

## @nczksv 2018-04-05 10:31:20

$$H_n = \frac{\binom{(n+1)!+n}{n}-1}{(n+1)!}-(n+1)\Biggl\lfloor \frac{\binom{(n+1)!+n}{n}-1}{(n+1)(n+1)!}\Biggr\rfloor $$

## @RocketNuts 2018-05-16 23:20:55

That's a remarkable closed form, do you have a source for a proof or more detailed elaboration on how this is derived?

## @Gerry Myerson 2011-07-20 06:11:56

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 Machinery28(1981) 305-350, DOI: 10.1145/322248.322255. Quoting,Undoubtedly the particular problem of $H_n$ goes back well before 1981. The references in Karr's paper may be of some help here.

## @J. M. is a poor mathematician 2011-07-20 11:20:45

For lazy people like me...

## @lhf 2011-07-21 22:10:40

There is also this paper: Michael Karr, "Theory of summation in finite terms",

Journal of Symbolic Computation1 (1985), no. 3, 303–315. MR0849038 (89a:12016)## @Jaume Oliver Lafont 2016-01-25 23:24:55

The following series shows the relationship between the harmonic numbers and the logarithm of odd integers.

$$ log(2n+1)=H_n+\sum_{k=1}^{\infty}\left(\sum_{i=-n}^{n}\frac{1}{(2n+1)k+i}-\frac{1}{k}\right) $$ https://math.stackexchange.com/a/1602945/134791

Equivalently, $$ H_n=log(2n+1)-\sum_{k=1}^{\infty}\left(\sum_{i=-n}^{n}\frac{1}{(2n+1)k+i}-\frac{1}{k}\right) $$

An integral form is given by

$$H_n=\log(2n+1)-\int_{0}^{1} \frac{x^n(1-x)}{\sum_{k=0}^{2n}x^k} \left( \frac{n(n+1)}{2}x^{n-1}+\sum_{k=0}^{n-2}\frac{(k+1)(k+2)}{2}\left(x^k+x^{2(n-1)-k}\right)\right)dx$$

An integral to prove that $\log(2n+1) \ge H_n$

## @Marc Paul 2016-01-25 23:48:24

I'm not sure that I would call those expresions 'closed-form'.

## @Jaume Oliver Lafont 2016-01-26 08:56:38

Yes, they should be regarded as analytic expression because they include infinite summation. Is that what you mean? However, this is the "closest-form" I am aware of for linking $H_n$ and $log(f(n))$

## @Vladimir Reshetnikov 2013-05-05 18:09:25

Harmonic numbers can be represented in terms of

integralsof elementary functions: $$H_n=\frac{\int_0^{\infty} x^n e^{-x} \log x \; dx}{\int_0^{\infty} x^n e^{-x} dx}-\int_0^{\infty} e^{-x} \log x \; dx.$$ This formula could also be used to generalize harmonic numbers to fractional or even complex arguments. These generalized harmonic numbers retain some of their useful properties, for example, $$H_z=H_{z-1}+\frac{1}{z}.$$## @Rob Bland 2015-11-12 02:17:24

The Harmonic numbers can also be represented by the integral $H_n = \int_0^1 \frac{1-x^n}{1-x} dx$

## @Peter Sheldrick 2011-07-20 09:21:48

This is probably not what you were looking for (since it isn't in terms of rational or elementary functions), but for the harmonic numbers we have

$$H_n=\frac{1}{n!}\left[{n+1 \atop 2}\right]$$

where $\left[{n \atop k}\right]$ are the (unsigned) Stirling numbers of the first kind (page 261 from the book

Concrete Mathematicsby Graham, Knuth and Patashnik - second edition).For the generalized harmonic numbers I like this formula - even though it does involve an integral and Riemann zeta...

Maybe you prefer this