The Sum of the Approximation Errors of Harmonic Numbers

Appendix A The Sum of the Approximation Errors of Harmonic Numbers

A.1 Introduction

In this appendix, we compile a list of identities that involve the sum of approximation errors of harmonic numbers by the natural logarithmic function. Speciﬁcally, a famous result, due to Euler, is that the limit:

˚ Xn 1 lim log n n!1 k kD1 exists and is given by a constant D 0:55721. This was proved in Chap. 2. Consequently, we know that the following alternating series:

X1 k .1/ Hk log k (A.1.1) kD1 converges to some ﬁnite value. Our ﬁrst goal is to derive an exact expression to this value. Using the Euler-Maclaurin summation formula, it can be shown that the approximation error Hn log n has the asymptotic expansion:

1 1 X B H log n k n 2n knk kD2 P 1 Œ This shows that the sum of approximation errors kD1 Hk log k diverges. Adding an additional term from the asymptotic expansion above gives us: 1 H log n O.1=n2/ n 2n

Hence, the following series:

X1 1 H log k (A.1.2) k 2k kD1 converges to some ﬁnite value. Our second goal is to derive a closed-form expression for this series. Finally, using the Euler-Maclaurin summation formula, Cesaro and Ramanujan, among others, deduced the following asymptotic expansion [Seb02]: p 1 H log n.n C 1/ n 6n.n C 1/

As a result, the natural logarithms of the geometric means offer a more accurate approximation to the harmonic numbers. In particular, the following series:

X1 p Hk log k.k C 1/ (A.1.3) kD1 converges to some ﬁnite value. Our third goal is to derive a closed-form expression for this series.

A.2 Main Theorem

Remarkably, for all the three series in Eqs. A.1.1ÐA.1.3, their exact values are expressed in terms of itself as well as log . More speciﬁcally, we have the following theorem: Theorem A.1 The sum of approximation errors of harmonic numbers by the natural logarithmic function satisfy the following identities:

1 h i X 1 log.2/ 1 log.k/ C H C D k 2k 2 kD1 1 h i X p log.2/ 1 k.k C 1/ C H D log k 2 kD1 1 h i X log .1/k k C H D log k 2 kD1

Proof The third identity was proved in Eq. 5.1.33 using the theory of summability of divergent series. The second equation was proved in Eq. 6.2.12 using the inﬁnite Reference 153 product representation of the superfactorial function. The ﬁrst identity follows from the second identity because:

X1 1 X1 p 11 1 log k C H C D log k.k C 1/ C H C log.1 C / k 2k k 2 k k kD1 kD1 .2/ 1 D log C 2 2 .2/ 1 D log 2

Reference

[Seb02] P. Sebha, Collection of formulae for Euler’s constant (2002). http://scipp.ucsc.edu/~haber/ archives/physics116A06/euler2.ps. Retrieved on March 2011 Glossary

P 1 ˛ Abel Summability Method Given a series kD0 k, the Abel summability method assigns the following value if the limit exists:

X1 X1 k ˛k , lim ˛k x x!1 kD0 kD0

The Abel summability method is consistent with, but is more powerful than, all Nörlund Means. Bernoulli Numbers Bernoulli numbers is a sequence of rational numbers that arise frequently in number theory. Their ﬁrst few terms are given by: 1 1 1 B0 D 1; B1 D ; B2 D ; B3 D 0; B4 D ; B5 D 0;::: 2 6 30

D1 Throughout this monograph, we adopt the convention B1 2 . Bernoulli numbers vanish at positive odd integers exceeding 1. The generating function of Bernoulli numbers is given by:

1 x X B D r xr ex 1 rŠ rD0

Composite Finite Sums A function f .n/ is a composite ﬁnite sum if it is of the form:

Xn1 f .n/ D g.k; n/ kD0

PHere, the iterated functionP g.k; n/ can depend on the bound n. The functions n1 .1 C = / n1 . C /1 kD0 log k n and kD0 k n are examples to composite ﬁnite sums. Glaisher Approximation The Glaisher approximation is an asymptotic expression to the hyperfactorial function. It states that:

2 H.n/ Aen =4 nn .nC1/=2C1=12; with a ratio that goes to unity as n !1. Here, A 1:2824 is often referred to as the Glaisher-Kinkelin constant. Hyperfactorial Function The hyperfactorial function H.n/ is deﬁned for n 2 N by the formula:

Yn H.n/ D kk kD1

LindelöfP Summability Method The Lindelöf summability method assigns to a 1 ˛ series kD0 k the value: X1 X1 ı k ˛k , lim k ˛k ı!0 kD0 kD0

It can correctly sum any Taylor series expansion in the Mittag-Lefﬂer star. Nearly Convergent Functions A function f .x/ is called nearly convergent if 0 limx!1 f .x/ D 0 and one of the following two conditions holds: 1. f .x/ is asymptotically non-decreasing and concave. More precisely, there exists 0 .2/ x0 such that for all x > x0,wehavef .x/ 0 and f .x/ 0. 2. f .x/ is asymptotically non-increasing and convex. More precisely, there exists x0 0 .2/ such that for all x > x0,wehavef .x/ 0 and f .x/ 0 p Examples of nearly convergent functions include x, 1=x,andlogx. Polynomial Order For any function f .x/, its polynomial order is the minimum r 0 such that:

drC1 lim f .x/ D 0 x!1 d xrC1 p . / D For example,p the function f x x x has a polynomial order of 1 because the ﬁrst 3 x !1 p3 derivative 2 does not vanish as x whereas its second derivative 4 x does. Glossary 157

Riemann Zeta Function The Riemann zeta function .s/ is deﬁned for R.s/>1 by the series:

X1 1 .s/ D ks kD1

For positive even integers, Euler showed that the following equation holds:

2s B2 .2/ .2s/ D .1/sC1 s 2.2s/Š

Semi-linear Simple Finite Sums A functionP f .n/ is a semi-linear simple finite sum n1 . / . / if it is a simple finite sum of the form kD0 g k and g k is a nearly-convergent function. Simple Finite Sums A function f .n/ is a simple finite sum if it is of the form:

Xn1 f .n/ D g.k/ kD0

Here, theP iterated function g.k/ is independent of n. ForP example, the log-factorial n1 .1 C / n1 1 function kD0 log k and the harmonic number kD0 1Ck are simple finite sums. Stirling Approximation The Stirling approximation is an asymptotic expression for the factorial function. It states that: p n n nŠ 2n ; e with a ratio that goes to unity as n !1. Summability Methods A summability method is a method of defining divergent series. One simple summability method is to define a series by the following expression:

X1 1 Xn XnC1 ˛k , lim ˛k C ˛k n!1 2 kD0 kD0 kD0

1 This deﬁnition assigns the value 2 to the Grandi series 11C11C1:::,which agrees with other methods. Two summability methods are called consistent if they always assign the same value to a series whenever the series is well-deﬁned under both methods. A summability method T1 is stronger than a summability method T2 if both are consistent and T1 can sum more series. 158 Glossary

Three desirable properties of summability methods are: 1. Regularity: A summability method is called regular if it agrees with ordinary summation whenever a series converges in the classical sense. 2. Linearity: A summability method is linear if we always have:

X1 X1 X1 ˛i C ˇi D ˛i C ˇi i i i

3. Stability: A summability method is called stable if we always have:

X1 X1 ˛k D ˛0 C ˛1Ck kD0 kD0

Superfactorial Function The superfactorial function S.n/ is deﬁned for n 2 N by the formula:

Yn S.n/ D kŠ kD1 P n1 . / Telescoping Sums and Products A simple ﬁnite sum of the form kD0 g k is called telescoping if it can be written in the form:

Xn f .n/ D h.k/ h.k C 1/ kDa

An often-cited example is the ﬁnite sum:

Xn 1 f .n/ D n .n C 1/ kD1

This ﬁnite sum can be rewritten as:

Xn 1 Xn 1 1 f .n/ D D k .k C 1/ k k C 1 kD1 kD1 1 1 1 1 1 1 1 D 1 Œ Œ ::: Œ 2 2 3 3 n n n C 1 1 D 1 n C 1 Q . / D n1 . / Hence, the sum is telescoping. A ﬁnite product f n kD0 g k is called telescoping if log f .n/ is a telescoping sum. Glossary 159

The Bohr-Mollerup Theorem The Bohr-Mollerup theorem is one possible char- acterization for the uniqueness of the gamma function in generalizing the deﬁnition of the discrete factorial function. The theorem is due to Harald Bohr and Johannes Mollerup who proved that .n C 1/ is the only function that satisﬁes: 1. f .n/ D nf.n/ 2. f .1/ D 1 3. f .n/ is logarithmically convex.

The Cesàro Means The Cesàro mean is a summability method introducedP by 1 ˛ Ernesto Cesàro. It is an averaging technique, which deﬁnes a series kD0 k by:

X1 ˚1 Xn Xj ˛k , lim ˛k n!1 n kD0 jD1 kD0 P j ˛ !1 For convergent sums, kD0 k tends to a limit V as j so the overall Cesàro mean tends to the same limit. Therefore, the Cesàro mean is a regular summability method. It is also linear and stable. The Cesàro mean is occasionally referred to as the Cesàro summability method. P 1 .1/k˛ The Euler Sum Given a series kD0 k, its Euler sum is deﬁned by the limit:

X1 Xn .1/k k k .1/ ˛k , lim ˛; n!1 2kC1 kD0 kD0 if the limit exists. The Euler sum usually agrees with the T definition of series whenever both are defined. The Gamma Function The gamma function, denoted by the Greek capital letter , is a generalization to the discrete factorial function. It is defined for all complex numbers except the negative integers and zero. It satisfies: Z 1 1 Y1 1 C 1 n . / D n1 t D k D . 1/Š n t e d t n n 0 n 1 C kD1 k

Both the gamma function and its inﬁnite product representation are due to Euler.

The Harmonic Numbers The harmonic numbers Hn are given by:

Xn1 1 H D n 1 C k kD0 160 Glossary

. / D d . / If we let n denotes the digamma function, where d x log x ,then harmonic numbers can be extended to the complex plane using:

Hn D .n C 1/ C

Here, 0:5772 is Euler’s constant. Asymptotically, we have:

Hn C log n; with an error that goes to zero as n !1. The Mittag-Lefﬂer Star The Mittag-Lefﬂer star of a function f .x/ around the origin is the set of all points z 2 C such that the line segment [0, z] does not pass through a singularity point of f .x/. For example, the star of the function log .1 C x/ includes the entire complex plane C except the negative real line .1; 1. The Nörlund Means The Nörlund means are a generalization to the Cesàro means. P pn Here, suppose pj is a sequence of positive terms that satisfy n ! 0. Then, the kD0 pk Nörlund mean of a sequence .s0; s1;:::/is given by: C CC pns0 pnP1s1 p0sn lim n n!1 kD0 pk

The limit of an inﬁnite sequence .s0; s1;:::/ is deﬁned by its Nörlund mean. All Nörlund means are regular, linear, and stable. The Sampling Theorem The Sampling Theorem is a fundamental result in information theory. It states that band-limited functions with bandwidth B can be perfectly reconstructed from their discrete samples if the sampling rate is larger than 2B. The requirement that the sampling rate exceeds twice the bandwidth is called the Nyquist criteria.

The T Definition of Series ThePT definition is a generalization to the classical 1 . / definition of series. Given a series kD0 g k ,let: X1 h.z/ D g.k/ zk (1) kD0

In other words, h.z/ is the function whose Taylor series expansion around the origin isgivenbyEq.P 1.Ifh.z/ is analytic in the domain z 2 Œ0; 1, then the T value of the 1 .1/ 1 2 C 3 4 C ::: series kD0 ak is defined by h . For example, the infinite sum 1 T is assigned the value 4 by because it arises out of the Taylor series expansion of f .x/ D .1 C x/2 at x D 1. The definition T is regular, linear, and stable.

The T Sequence Limit The T sequence limit of an infiniteP sequence S D . ; ; ;:::/ T C 1 s0 s1 s2 is defined by the -value of the infinite sum: s0 kD0 sk. Glossary 161

The Summability Method TheP summability method , which we introduced in 1 ˛ this monograph, assigns to a series kD0 k the value:

1 X Xn Yk j 1 ˛k , lim n.k/˛k; where n.k/ D 1 n!1 n kD0 kD0 jD1

Triangular Numbers Triangular numbers is the sequence of integers 1; 3; 6; 10; : : : whosePkth element is given by k.k C 1/=2. They are generated by the simple ﬁnite n1 . C 1/ sum kD0 k . Index

summability method, 16, 17, 47, 50, 75, 98, ceiling function, 116 103, 120, 124, 128, 133, 136, 161 Cesàre means, 11, 16, 66, 71, 103, 159 T sequence limit, 70, 160. see also generalized Cesàro summability method. see Cesàro means definition T chain rule, 56 composite finite product, 12, 55 composite finite sum, 12, 55, 100, 129, 155 Abel summability method, 12, 71, 75, 155 convergence acceleration, 98, 106, 124 Academia Algebrae, 28 cook-book recipe, 28. see also power sum alternating power sum, 97 alternating sum, 94, 97, 98, 100 analytic continuation, 1, 49, 72, 99, 144 DFT. see discrete Fourier transform Aryabhata, 14 digamma function, 13, 37, 146, 160. see also Aryabhatiya, 14 harmonic numbers asymptotic analysis, 15, 42, 47, 58, 93, 98, discrete Fourier transform, 106 106, 125, 142 discrete function, 5, 13 asymptotic differentiation order, 47, 97, 105, discrete harmonic function, 116 116, 121, 124, 125, 127, 129, 156 divergent series, 10, 16, 67, 93. see also summability theory double gamma function, 14. see also backward difference operator, 134 superfactorial bandlimited function, 137 bandwidth, 137Ð139 Barnes G-function, 14. see also superfactorial Euler constant, 6, 30, 35, 37, 124, 146, 160 Berndt-Schoenfeld periodic analogs, 15, 93 Euler polynomials, 95 Bernoulli numbers, 4, 24, 60, 73, 96, 155, 157 Euler summation, 74, 99, 100, 159 Bernoulli-Faulhaber formula, 4, 8, 27, 49, 97, Euler’s generalized constants, 30 116, 125 Euler-Macluarin summation formula, 15, 45, beta function, 39 47, 50, 93, 117, 124, 141 Bohr-Mollerup theorem, 4, 36, 159 Boole summation formula, 15, 93, 95 factorial function, 1Ð4, 6, 13, 16, 28, 35Ð37, 39, 42, 59, 89, 119, 124, 159. see Cauchy numbers, 147 also gamma function Cauchy product, 68 falling factorial, 141 Cauchy repeated integration formula, 58 finite differences, 15, 133, 134, 143

© Springer International Publishing AG 2018 163 I. M. Alabdulmohsin, Summability Calculus, https://doi.org/10.1007/978-3-319-74648-7 164 Index forward difference operator, 134 Newton interpolation formula, 17, 136, 141 Fourier transform, 137, 138 numerical integration, 14, 60, 145. see also Riemann sum Nyquist criteria, 137. see also sampling gamma function, 4, 13, 17, 36, 37, 119, 124, theorem, 161 159. see also factorial function generalized definition T, 67, 160 generalized harmonic numbers, 13, 145 oscillating sums, 102, 106, 107, 110, 117, 126, generating function method, 12. see also Abel 127, 142 summability method geometric series, 8, 14 Glaisher approximation, 6, 16, 42, 43, 156. see PI function, 13, 37 also hyperfactorial polynomial fitting, 8, 15, 17, 47, 125 Glaisher’s constant, 6, 156 polynomial order, 13 Grandi series, 10, 66, 102 power sum, 3, 4, 8, 27, 49 Gregory coefficients, 141, 145 Gregory quadrature formula, 14, 15, 141, 145 Ramanujan, 16, 61, 107 regular paper-folding sequence, 73 Hadamard’s factorial function, 37 regularity, 68, 80, 158 half sampling theorem, 17. see also sampling Riemann sum, 60, 141. see also numerical theorem integration harmonic numbers, 1, 5, 6, 16, 35, 53, 57, 124, rigidity, 8, 68, 69 145, 159. see also digamma function Hasse, Helmut, 75, 145 Hutton summability method, 66 sampling theorem, 8, 17, 137, 139, 161 hyperfactorial, 5, 6, 16, 42, 123, 156 second factorials, 98 hypergeometric series, 144 semi-linear simple finite sums, 29, 35, 117, 119, 157 Ser formula, 144 interpolation, 2, 3, 8, 23, 37, 47, 120, 125 Shannon-Nyquist sampling theorem. see sampling theorem sign sequence, 104Ð106 Kluyver formula, 146 simple finite product, 12, 45 simple finite sum, 12, 45, 94, 102, 105, 107Ð110, 115, 117, 119, 121, 126, Legendre’s normalization, 14 134, 141, 143, 157 Lerch’s Transcendent, 144 stability, 68, 158 Lindelöf summability method, 68, 72, 75 Stieltjes constants, 30 Lindelöf summmability method, 137, 156 Stirling approximation, 6, 16, 42, 49, 61, 98, log-factorial function, 13, 35, 36, 42, 50, 119 157 log-superfactorial function, 51, 52, 123, 144 Stirling numbers of the first kind, 142 Stirling numbers of the second kind, 73 successive polynomial approximation method, Müller-Schleicher method, 17, 115, 117 24 matrix summability, 80 summability theory, 10, 16, 65, 108, 142, 157. Mittag-Leffler star, 68, 137, 160 see also divergent series Mittag-Leffler summability method, 66, 68, 75 summation by parts, 53 multiresolution, 139 superfactorial, 5, 14, 16, 50, 52, 123, 144, 158

Nörlund means, 71, 160 Taylor series, 7, 31, 59, 67, 68, 75, 124, 137 nearly-convergent function, 29, 156 telescoping sums, 32, 158 Index 165

Toeplitz-Schur Theorem, 80 Wallis formula, 61 totally regular, 72 wavelet sampling, 139 triangular numbers, 1Ð4, 27. see also power Weierstrass theorem, 23 sum, 161

zeta function, 14, 35, 74, 145, 157 undersampling, 137, 143