# Proving that $~\sum\limits_{k=2}^{\infty}\frac{(-1)^k}{k^2}~H_k~H_{k-1}=\frac{3}{16}~\zeta(4)$

To show that $$\sum\limits_{k=2}^{\infty} \frac{(-1)^{k}}{k^{2}} \, \left(1+\frac{1}{2}+...+\frac{1}{k}\right) \cdot \left(1+\frac{1}{2}+...+\frac{1}{k-1}\right) = \frac{3}{16}\zeta(4).$$

I came across this when trying to solve a problem from the current edition of the American Mathematical Monthly. Is there some easy way to show this? I checked numerically that this series does converge to the value of $\frac{3}{16}\zeta(4)$.

Note: An alternate form, with $H_{n}$ being the harmonic numbers, is: $$\sum\limits_{k=2}^{\infty} \frac{(-1)^{k}}{k^{2}} \, H_{k} \, H_{k-1} = \frac{3}{16}\zeta(4).$$

• algo.inria.fr/flajolet/Publications/FlSa98.pdf has several techniques that might be useful, but I haven't finished reading it. Either way, it's a fun read :D Jul 28, 2017 at 20:30
• My go-to reference for this kind of stuff are the Borwein brothers. Check out this paper: ams.org/journals/proc/1995-123-04/S0002-9939-1995-1231029-X/… specifically, the theorem and its corollaries under "Main Results" on page 2. My money is on manipulating the integral to give you that $H_kH_{k-1}$ combination. In particular check out the integrals that follow under equation (16) on page 7, which give the non-alternating sum version of your identity. Jul 28, 2017 at 20:48
• Link for the AMM problem. Here's an old spoiler blog post. But trust me, the problem in that form is much simpler than what I did in that blog. :P
– r9m
Jul 28, 2017 at 21:56
• @r9m impressive work! Jul 28, 2017 at 23:33
• @Alex Meiburg,@ Alex R: Thanks a lot for all the references, they are all very interesting. The link zerocollar.blogspot.in/2015/05/… shared by r9m pretty much solves it. Jul 29, 2017 at 2:51

## 3 Answers

This is an opportunity to make a tribute to Pieter J. de Doelder (1919-1994) from Eindhoven University of Technology, who evaluated in closed form the given series in a somewhat famous paper (p. 132-133 2.3) (1991). One may start by using the following identity coming from the Cauchy product, $$\ln^2(1+x) =2\sum_{n=1}^\infty (-1)^{n-1} \frac{H_n}{n+1} \:x^{n+1}$$ giving $$\int_0^1\frac{\ln(1-x)\ln^2(1+x)}{x} \:dx=2\sum_{n=1}^\infty (-1)^{n-1} \frac{H_n}{n+1} \:\int_0^1 x^{n}\ln(1-x)\:dx,$$ then using the standard evaluation $$\int_0^1 x^{n}\ln(1-x)\:dx =-\frac{H_{n+1}}{n+1},\quad n\ge0,$$ one gets

$$\int_0^1\frac{\ln(1-x)\ln^2(1+x)}{x}\:dx=2\sum\limits_{n=2}^{\infty} (-1)^{n-1} \frac{H_n H_{n-1}}{n^{2}}. \tag1$$

Here are the main steps which de Doelder took to evaluate the related integral.

We clearly have \begin{align} \int_0^1\ln^3\left(\frac{1+x}{1-x}\right)\:\frac{dx}{x}&=\int_0^1\frac{\ln^3\left(1+x\right)}{x}\:dx-3\int_0^1\frac{\ln(1-x)\ln^2(1+x)}{x}\:dx \\\\&+3\int_0^1\frac{\ln^2(1-x)\ln(1+x)}{x}\:dx-\int_0^1\frac{\ln^3\left(1-x\right)}{x}\:dx \end{align} and \begin{align} \int_0^1\frac{\ln^3\left(1-x^2\right)}{x}\:dx&=\int_0^1\frac{\ln^3\left(1+x\right)}{x}\:dx+3\int_0^1\frac{\ln(1-x)\ln^2(1+x)}{x}\:dx \\\\&+3\int_0^1\frac{\ln^2(1-x)\ln(1+x)}{x}\:dx+\int_0^1\frac{\ln^3\left(1-x\right)}{x}\:dx, \end{align} substracting the two equalities, \begin{align} 6\!\!\int_0^1\!\frac{\ln(1-x)\ln^2(1+x)}{x}dx&=\!\int_0^1\!\frac{\ln^3\left(1-x^2\right)}{x}dx-\!\int_0^1\!\!\ln^3\left(\frac{1+x}{1-x}\right)\frac{dx}{x}-2\!\int_0^1\!\frac{\ln^3\left(1-x\right)}{x}dx \\\\&=I_1-I_2-2I_3. \end{align} It is easy to obtain \begin{align} I_1=\int_0^1\!\frac{\ln^3\left(1-x^2\right)}{x}dx&=\frac12 \int_0^1\!\frac{\ln^3\left(1-u\right)}{u}du \quad (u=x^2) \\&=\frac12 \int_0^1\!\frac{\ln^3 v}{1-v}dv \quad (v=1-u) \\&=\frac12 \sum_{n=0}^\infty \int_0^1\!v^n\ln^3 v\:dv \\&=-3\sum_{n=1}^\infty \frac1{n^4} \\&=-\frac{\pi^4}{30}, \end{align} similarly \begin{align} I_3=\int_0^1\!\frac{\ln^3\left(1-x\right)}{x}dx=-\frac{\pi^4}{15}. \end{align} By the change of variable, $u=\dfrac{1-x}{1+x}$, one has $\dfrac{dx}{x}=\dfrac{-2\:du}{1-u^2}$ getting \begin{align} I_2=\int_0^1\!\!\ln^3\left(\frac{1+x}{1-x}\right)\frac{dx}{x}&=-2\int_0^1\!\frac{\ln^3 u}{1-u^2}du \\&=-2\sum_{n=0}^\infty \int_0^1\!u^{2n}\ln^3u\:dv \\&=12\sum_{n=0}^\infty \frac1{(2n+1)^4} \\&=\frac{\pi^4}{8}. \end{align} Then,

$$\int_0^1\frac{\ln(1-x)\ln^2(1+x)}{x}\:dx=-\frac{\pi^4}{240} \tag2$$

and

$$\sum\limits_{n=2}^{\infty} (-1)^{n} \frac{H_n H_{n-1}}{n^{2}}= \frac{3}{16}\zeta(4)=\frac{\pi^4}{480},\tag3$$

as announced.

• thanks! This integral is exactly the problem proposed in AMM this month. Would it be wrong to publicly keep this solution up here in that case? Jul 29, 2017 at 12:39
• (+1) Nice solution .. !! :) Elementary and elegant as usual!
– r9m
Jul 29, 2017 at 18:25
• @Aritro Pathak You are welcome! It's no big deal, this monthly problem is not original, it is known for years. Jul 29, 2017 at 18:25
• Ingenious! Regarding the question of motivation for the solution, I suppose you do have to memorize identities like the first and third, is that right?
– Hans
Jul 29, 2017 at 23:06
• @Hans Right, and if you don't memorize it, you know it does exist somewhere. Thanks. Jul 30, 2017 at 0:31

Although I have seen too few proofs in this field to be able to compare, this approach might be interesting.

We transform the sum to a fourfold integral which Mathematica can solve immediately. I hope it should be possible to solve the integral "mathematically" as well, which would then complete the proof.

We have to calculate

$$s=\sum _{k=2}^{\infty } \frac{(-1)^k}{k^2} H(k) H(k-1)$$

Writing

$$\frac{1}{n^2}=\int_0^1 \frac{1}{x}\,dx \int_0^x y^{n-1} \, dy$$

$$\frac{1}{n}=\int_0^1 r^{n-1} \, dr$$

and

$$H(k)=\sum _{n=1}^k \frac{1}{n}=\int_0^1 \left(\sum _{n=1}^k r^{n-1}\right) \, dr=\int_0^1 \frac{1-r^k}{1-r} \, dr$$

the sum $s$ below the integrals becomes

$$si=\frac{1}{x(1-r)(1-s)}\sum _{k=2}^{\infty } (-1)^k \left(1-r^k\right) \left(1-s^{k-1}\right) y^{k-1}$$

Which evaluates to

$$si = \frac{y \left(r^2 s^2 y+r^2 s-r^2 y-r^2-r s^2 y+r y-s+1\right)}{(1-r) (1-s) x (y+1) (r y+1) (s y+1) (r s y+1)}$$

Now the integral to be evaluated is

$$s4 = \int _0^1 dx\int _0^x dy\int _0^1 dr\int _0^1 ds \; si$$

Mathematica finds immediately

$$s4 = \frac{\pi ^4}{480}$$

Since

$$\zeta (4)=\frac{\pi ^4}{90}$$

and

$$\frac{90}{480} = \frac{3}{16}$$

we have finally

$$s = \frac{3}{16} \zeta(4)$$

• Thank you for this point of view! (+1) Is it possible to have a screenshot of your instructions in Mathematica? Thanks. Jul 30, 2017 at 7:44
• @Olivier Oloa No screenshot necessary. Using the integrand $si$ as defined above it's just this statement Integrate[si, {x, 0, 1}, {y, 0, x}, {r, 0, 1}, {s, 0, 1}]. To my big surprise the result pi^4/480 was returned within a second. So if I had not known the result in advance I would have found it heuristically. This is not replacing a strict proof, of course. Jul 30, 2017 at 8:49
• Maple can't integrate, so it's not that trivial ;-( Jan 8, 2019 at 0:35
• @Diger I have never used maple but did you try different versions of maple? I keep several older versions of Mathematica because some are better in integrating than others. Jan 9, 2019 at 8:27
• Unfortunately not; just tried 2018. Jan 9, 2019 at 14:27

Bonus

$$S=\sum_{k=2}^\infty\frac{(-1)^k}{k^2}H_kH_{k-1}=\sum_{k=1}^\infty\frac{(-1)^k}{k^2}H_kH_{k-1}=\sum_{k=1}^\infty\frac{(-1)^kH_k^2}{k^2}-\sum_{k=1}^\infty\frac{(-1)^kH_k}{k^3}$$

Since $$S=\frac3{16}\zeta(4)$$ and $$\sum_{k=1}^\infty\frac{(-1)^kH_k}{k^3}$$ $$=2\operatorname{Li_4}\left(\frac12\right)-\frac{11}4\zeta(4)+\frac74\ln2\zeta(3)-\frac12\ln^22\zeta(2)+\frac{1}{12}\ln^42$$

then

$$\sum_{n=1}^{\infty}\frac{(-1)^kH_k^2}{k^2}=2\operatorname{Li}_4\left(\frac12\right)-\frac{41}{16}\zeta(4)+\frac74\ln2\zeta(3)-\frac12\ln^22\zeta(2)+\frac1{12}\ln^42$$