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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 approxima- tion 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 approx- imation 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 © Springer International Publishing AG 2018 151 I. M. Alabdulmohsin, Summability Calculus, https://doi.org/10.1007/978-3-319-74648-7 152 A The Sum of the Approximation Errors of Harmonic Numbers 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.1.1–A.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 © Springer International Publishing AG 2018 155 I. M. Alabdulmohsin, Summability Calculus, https://doi.org/10.1007/978-3-319-74648-7 156 Glossary 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/ Ä 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/ Ä 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: Á 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 Á ˛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.
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