An integral representation of nth Harmonic number is

$$H_n = \displaystyle \int_0^1 \frac{1 - x^n}{1 - x}\,dx$$

Wikipedia states that for every x > 0, integer or not, we have: $$H_{n} = n \displaystyle\sum_{k=1}^\infty \frac{1}{k(n+k)}$$

How can I get to this result from integral representation?


We have $$H_n = \sum_{k=1}^n \dfrac1k = \sum_{k=1}^n \int_0^1 x^{k-1} dx = \int_0^1 \left(\sum_{k=1}^n x^{k-1} \right) dx = \int_0^1 \left(\dfrac{1-x^n}{1-x} \right)dx$$

You may also want to look at these answers [1] and [2] and , where I derived a similar/same result. The second part of your question is answered in [1].

  • $\begingroup$ I have 2 questions about your proof. Q1) Does it work for all fixed z? (It was not stated for what z does it work.). Q2) How do you substantiate interchange of sum and integral signs in 2nd row? Thank you. $\endgroup$ – Karlis Olte Apr 7 '13 at 9:50
  • $\begingroup$ The integral is linear, so integral of (finite) sums is the (finite) sum of the integrals. $\endgroup$ – copper.hat Apr 7 '13 at 16:09
  • $\begingroup$ @copper.hat there is infinity sign above summation sign, doesn't it mean it is infinite sum? $\endgroup$ – Karlis Olte Apr 7 '13 at 16:59
  • $\begingroup$ I don't see any $\infty$ in the above answer? $\endgroup$ – copper.hat Apr 7 '13 at 17:06
  • 2
    $\begingroup$ @user64985 It might be better if you had asked this question in the comment section of reference [1], sicne this is related to that. Anyway, the idea is as follows. Since the convergence of the series is uniform in any closed interval of the form $[0,1-\epsilon]$, we are allowed to interchange the integral and summation for $\int_0^{1-\epsilon} f(x) dx$. Now argue that the function you get is continuous and hence the answer you can let $\epsilon \to 0$. $\endgroup$ – user17762 Apr 7 '13 at 17:27

For the second, $$n \sum\limits_{k=1}^{+ \infty} \frac{1}{k(n+k)}= \sum\limits_{k=1}^{+ \infty} \left( \frac{1}{k}-\frac{1}{n+k} \right)$$


"How can I get to this result from integral representation?"

Here's how, taking your request literally. The trick is integrating by parts

$$ \begin{align} &\int_0^1 \frac{1-x^n}{1-x} \, dx\tag{1}\\ &=-\int_0^1 \left(1-x^n\right) \frac{d}{dx} \log (1-x) \, dx\tag{2a}\\ &=-\left(1-x^n\right) \log (1-x)\Big|^{1}_{0}-\int_0^1 n\; x^{n-1} \log (1-x) \, dx\tag{2b}\\ &=\int_0^1 n\; x^{n-1} \sum _{k=1}^{\infty } \frac{x^k}{k} \, dx\tag{2c}\\ &= \sum _{k=1}^{\infty } \frac{n}{k}\int_0^1 x^{n+k-1} \, dx\tag{2d}\\ &=\sum _{k=1}^{\infty } \frac{n}{k (n+k)}\tag{2e}\\ \end{align} $$

(2a): rewrite the integrand to prepare integration by parts
(2b): do integration by parts. Notice that the integrated part vanishes at both ends of the interval
(2c): expand $\log (1-x)$ into a series
(2d): Exchange summation and integration
(2e): do the integral
Derivation finished


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