Journal ofClassical

Analysis

Volume 6, Number 2 (2015), 113–118 doi:10.7153/jca-06-09

REVIVING THE QUADRATIC SERIES OF AU–YEUNG

CORNEL IOAN VALEAN AND OVIDIU FURDUI

Abstract. In this paper we revive and bring to light the quadratic series of Au–Yeung

∞

∑n=1

(Hn

n

)2

=174

ζ (4),

where Hn denotes the n th harmonic number. We prove this series identity by using a techniquebased on the computation of a special logarithmic integral combined with Abel’s summationformula.

1. Introduction and the main result

The identity∞

∑n=1

(Hn

n

)2

=174

ζ (4) =17π4

360,

was discovered numerically by Enrico Au–Yeung an undergraduate student in the Fac-ulty of Mathematics in Waterloo and proved rigorously by David Borwein and JonathanBorwein in [3] who used Fourier series techniques combined with Parseval’s formulafor proving it. This quadratic series has become a classic in the theory of nonlinear har-monic series. We mention that a nonlinear harmonic series is a series which involvesproducts of at least two harmonic numbers [4]. It appears as a problem in [7, Problem2.6.1. p. 110], [5, Problem 3.70, p. 150] and a recent proof involving integrals ofpolylogarithm functions was given in [6].

In this paper we revive and bring to light the quadratic series of Au–Yeung bygiving a proof of it which is based on the calculation of a quadratic logarithmic integralcombined with Abel’s summation formula. We mention that our results are not newand they exist in the mathematical literature.

We state below the theorem we are going to prove.

THEOREM 1 (A quadratic series of Au–Yeung) The following equality holds:

∞

∑n=1

(Hn

n

)2

=174

ζ (4) =17π4

360,

where Hn is the nth harmonic number defined, for n � 1 , by Hn = 1+ 12 + · · ·+ 1

n .

Mathematics subject classification (2010): 40C10, 40A05.Keywords and phrases: Abel’s summation formula, logarithmic integrals, harmonic numbers,

quadratic series, Riemann zeta function.

c© � � , ZagrebPaper JCA-06-09

113

114 CORNEL IOAN VALEAN AND OVIDIU FURDUI

Before we prove Theorem 1 we collect some results we need in our analysis.

Recall that, Abel’s summation formula ([2, p. 55], [5, p. 258]) states that if

(an)n�1 and (bn)n�1 are two sequences of real numbers and An =n∑

k=1ak , then

n

∑k=1

akbk = Anbn+1 +n

∑k=1

Ak(bk −bk+1). (1)

We will also be using, in our calculations, the infinite version of the precedingformula

∞

∑k=1

akbk = limn→∞

(Anbn+1)+∞

∑k=1

Ak(bk −bk+1). (2)

Next we prove the following two lemmas which are used in the proof of Theorem1.

LEMMA 2 (Two logarithmic integrals and harmonic numbers) Let n � 1 be an integer.The following equalities hold:

(a)∫ 1

0xn−1 ln(1− x)dx = −Hn

n;

(b)∫ 1

0xn−1 ln2(1− x)dx =

2n

n

∑k=1

Hk

k=

H2n

n+

1n

(1+

122 + · · ·+ 1

n2

).

Proof. (a) We have,

∫ 1

0xn−1 ln(1− x)dx =

∫ 1

0xn−1

(∫ x

0− 1

1− tdt

)dx

= −∫ 1

0

11− t

(∫ 1

txn−1dx

)dt

= −1n

∫ 1

0

1− tn

1− tdt

= −1n

∫ 1

0

(1+ t + t2 + · · ·+ tn−1)dt

= −Hn

n.

REVIVING THE QUADRATIC SERIES OF AU-YEUNG 115

(b) We have,∫ 1

0xn−1 ln2(1− x)dx =

∫ 1

0xn−1

(∫ x

0−2

ln(1− t)1− t

dt

)dx

= −2∫ 1

0

ln(1− t)1− t

(∫ 1

txn−1dx

)dt

=−2n

∫ 1

0ln(1− t)

1− tn

1− tdt

= −2n

∫ 1

0ln(1− t)

(1+ t + t2 + · · ·+ tn−1)dt

=2n

n

∑k=1

Hk

k,

where the last equality follows based on part (a) of the lemma.It remains to prove that

H2n +1+

122 + · · ·+ 1

n2 = 2n

∑k=1

Hk

k.

It is worth mentioning that, the previous formula is known in the literature. Forexample, it appears in [1, Equation (3,62)] and it can be proved by mathematical induc-tion. However, we give below another proof of it which is based on the summation byparts formula. To see this, we use formula (1), with ak = 1

k and bk = Hk , and we getthat

n

∑k=1

Hk

k=

(1+

12

+ · · ·+ 1n

)Hn+1−

n

∑k=1

(1+

12

+ · · ·+ 1k

)1

k+1

= HnHn+1−n

∑k=1

Hk+1− 1k+1

k+1

= HnHn+1−n

∑k=1

Hk+1

k+1+

n

∑k=1

1(k+1)2

= HnHn+1−n+1

∑m=2

Hm

m+

n+1

∑m=2

1m2

= HnHn+1−n

∑m=1

Hm

m− Hn+1

n+1+

n

∑m=1

1m2 +

1(n+1)2

= H2n +1+

122 + · · ·+ 1

n2 −n

∑m=1

Hm

m,

and the lemma is proved.

LEMMA 3 (A special harmonic sum) The following equality holds:

∞

∑n=1

1n2

(1+

122 + · · ·+ 1

n2

)=

74

ζ (4).

116 CORNEL IOAN VALEAN AND OVIDIU FURDUI

Proof. We apply Abel’s summation formula (2), with an = 1n2 and bn = 1+ 1

22 +· · ·+ 1

n2 , and we have

S =∞

∑n=1

1+ 122 + · · ·+ 1

n2

n2

= limn→∞

(112 +

122 + · · ·+ 1

n2

)(1+

122 + · · ·+ 1

(n+1)2

)

−∞

∑n=1

112 + 1

22 + · · ·+ 1n2

(n+1)2

= ζ 2(2)−∞

∑n=1

1+ 122 + 1

32 + · · ·+ 1(n+1)2

(n+1)2 +∞

∑n=1

1(n+1)4

= ζ 2(2)−∞

∑m=2

1+ 122 + 1

32 + · · ·+ 1m2

m2 +∞

∑m=2

1m4

= ζ 2(2)−S+ ζ (4)

=72

ζ (4)−S.

We used that ζ 2(2) = 52 ζ (4) since ζ (2) = π2

6 and ζ (4) = π4

90 [9, p. 605].

The summation identity in Lemma 3 is known in the literature. It is a special caseof the following series formula involving the generalized harmonic number (see [10,Formula (1.1)]):

∞

∑n=1

H(p)n

np =12

(ζ 2(p)+ ζ (2p)

).

Now we are ready to prove Theorem 1.

Proof. We have, based on part (b) of Lemma 2, that

∫ 1

0

xn−1

nln2(1− x)dx =

H2n

n2 +1n2

(1+

122 + · · ·+ 1

n2

),

and it follows that

∞

∑n=1

∫ 1

0

xn−1

nln2(1− x)dx =

∞

∑n=1

H2n

n2 +∞

∑n=1

1n2

(1+

122 + · · ·+ 1

n2

). (3)

Using Tonelli’s theorem [8, p. 309] we are allowed to bring the sum under the

REVIVING THE QUADRATIC SERIES OF AU-YEUNG 117

integral and we have

∞

∑n=1

∫ 1

0

xn−1

nln2(1− x)dx =

∫ 1

0

∞

∑n=1

xn−1

nln2(1− x)dx

= −∫ 1

0

ln3(1− x)x

dx

= −∫ 1

0

ln3 y1− y

dy

= −∫ 1

0ln3 y

∞

∑k=0

ykdy

= −∞

∑k=0

∫ 1

0yk ln3 ydy

= 6∞

∑k=0

1(k+1)4

= 6ζ (4),

(4)

since∫ 10 yk ln3 ydy = − 6

(k+1)4 . Combining (3), (4) and Lemma 3 we get that

∞

∑n=1

(Hn

n

)2

=174

ζ (4),

and the theorem is proved.

RE F ER EN C ES

[1] H. ALZER, D. KARAYANNAKIS, H. M. SRIVASTAVA, Series representations of some mathematicalconstants, J. Math. Anal. Appl. 320, (2006) 145–162.

[2] D. D. BONAR, M. J. KOURY, Real Infinite Series, MAA, Washington DC, 2006.

[3] D. BORWEIN, J. M. BORWEIN, On an intriguing integral and some series related to ζ (4) , Proc. Am.Math. Soc. 123, (1995) 1191–1198.

[4] J. CHOI, H. M. SRIVASTAVA,Explicit evaluations of Euler and related sums, Ramanujan J. 10 (2005)51–70.

[5] O. FURDUI, Limits, Series and Fractional Part Integrals. Problems in Mathematical Analysis,Springer, New York, 2013.

[6] O. FURDUI, Series involving products of two harmonic numbers, Math. Mag. 84 (2011) 371–377.

[7] T-L. T. RADULESCU, V. D. RADULESCU, T. ANDREESCU, Problems in Real Analysis: AdvancedCalculus on the Real Axis, Springer, New York, 2009.

[8] H. L. ROYDEN, Real Analysis, Third Edition, Prentice Hall, Englewood Cliffs, NJ, 1988

118 CORNEL IOAN VALEAN AND OVIDIU FURDUI

[9] NIST Handbook of Mathematical Functions, Edited by Frank W.J. Olver, Daniel W. Lozier, Ronald F.Boisvert, Charles W. Clark, NIST National Institute of Standards and Technology US Department ofCommerce and Cambridge University Press, 2010.

[10] A. SOFO, Harmonic numbers of order two, Miskolc Math. Notes, 13 (2012) 499–514.

(Received September 4, 2014) Cornel Ioan ValeanTeremia Mare, Nr. 632

Timis, 307405Romania

e-mail: cornel2001 ro@yahoo.com

Ovidiu FurduiDepartment of Mathematics

Technical University of Cluj-NapocaStr. Memorandumului Nr. 28

400114, Cluj-Napoca, Romaniae-mail: Ovidiu.Furdui@math.utcluj.ro, ofurdui@yahoo.com

Journal of Classical Analysiswww.ele-math.comjca@ele-math.com