Editing List of logarithmic identities (section)

=== Harmonic number difference ===

It is not uncommon in advanced mathematics, particularly in [[analytic number theory]] and [[asymptotic analysis]], to encounter expressions involving differences or ratios of [[Harmonic number|harmonic numbers]] at scaled indices.<ref name="Flajolet2009AnalyticCombinatorics">{{cite book
 | last1 = Flajolet
 | first1 = Philippe
 | last2 = Sedgewick
 | first2 = Robert
 | title = Analytic Combinatorics
 | year = 2009
 | publisher = Cambridge University Press
 | isbn = 978-0521898065
 | page = 389
}} See page 117, and VI.8 definition of shifted harmonic numbers on page 389
</ref> The identity involving the limiting difference between harmonic numbers at scaled indices and its relationship to the logarithmic function provides an intriguing example of how discrete sequences can asymptotically relate to [[Continuous function|continuous functions]]. This identity is expressed as<ref name="Deveci2022DoubleSeries">{{cite arXiv
|last=Deveci
|first=Sinan
|title=On a Double Series Representation of the Natural Logarithm, the Asymptotic Behavior of Hölder Means, and an Elementary Estimate for the Prime Counting Function
|year=2022
|eprint=2211.10751
|class=math.NT
}} See Theorem 5.2. on pages 22 - 23</ref>

:<math>\lim_{{k \to \infty}} (H_{{k(n+1)}} - H_k) = \ln(n+1)</math>

which characterizes the behavior of harmonic numbers as they grow large. This approximation (which precisely equals <math>\ln(n+1)</math> in the limit) reflects how summation over increasing segments of the harmonic series exhibits [[Harmonic number#Calculation|integral properties]], giving insight into the interplay between discrete and continuous analysis. It also illustrates how understanding the behavior of sums and series at large scales can lead to insightful conclusions about their properties. Here <math>H_k</math> denotes the <math>k</math>-th harmonic number, defined as

:<math>H_k = \sum_{{j=1}}^k \frac{1}{j}</math>

The harmonic numbers are a fundamental sequence in number theory and analysis, known for their logarithmic growth. This result leverages the fact that the sum of the inverses of integers (i.e., harmonic numbers) can be closely approximated by the natural logarithm function, plus a [[Euler's constant|constant]], especially when extended over large intervals.<ref>{{cite book
 | last1 = Graham
 | first1 = Ronald L.
 | last2 = Knuth
 | first2 = Donald E.
 | last3 = Patashnik
 | first3 = Oren
 | title = Concrete Mathematics: A Foundation for Computer Science
 | year = 1994
 | publisher = Addison-Wesley
 | isbn = 0-201-55802-5
 | page = 429
}}
</ref><ref name="Flajolet2009AnalyticCombinatorics"/><ref>{{cite web
 | url = http://mathworld.wolfram.com/HarmonicNumber.html
 | title = Harmonic Number
 | publisher = Wolfram MathWorld
 | access-date = 2024-04-24
}} See formula 13.
</ref> As <math>k</math> tends towards infinity, the difference between the harmonic numbers <math>H_{k(n+1)}</math> and <math>H_k</math> converges to a non-zero value. This persistent non-zero difference, <math>\ln(n+1)</math>, precludes the possibility of the harmonic series approaching a finite limit, thus providing a clear mathematical articulation of its divergence.<ref>{{cite report
 | last1 = Kifowit
 | first1 = Steven J.
 | title = More Proofs of Divergence of the Harmonic Series
 | publisher = Prairie State College
 | year = 2019
 | url = https://stevekifowit.com/pubs/harm2.pdf
 | access-date = 2024-04-24
}} See Proofs 23 and 24 for details on the relationship between harmonic numbers and logarithmic functions.
</ref><ref>{{cite journal
 | last1 = Bell
 | first1 = Jordan
 | last2 = Blåsjö
 | first2 = Viktor
 | title = Pietro Mengoli's 1650 Proof That the Harmonic Series Diverges
 | journal = Mathematics Magazine
 | volume = 91
 | issue = 5
 | year = 2018
 | pages = 341–347
 | doi = 10.1080/0025570X.2018.1506656
 | jstor = 48665556
 | hdl = 1874/407528
 | url = https://www.jstor.org/stable/48665556
 | access-date = 2024-04-24
| hdl-access = free
 }}
</ref> The technique of approximating sums by integrals (specifically using the [[Integral test for convergence|integral test]] or by direct integral approximation) is fundamental in deriving such results. This specific identity can be a consequence of these approximations, considering:

:<math>\sum_{{j=k+1}}^{k(n+1)} \frac{1}{j} \approx \int_k^{k(n+1)} \frac{dx}{x}</math>

==== Harmonic limit derivation ====

The limit explores the growth of the harmonic numbers when indices are multiplied by a scaling factor and then differenced. It specifically captures the sum from <math>k+1</math> to <math>k(n+1)</math>:

:<math>H_{{k(n+1)}} - H_k = \sum_{{j=k+1}}^{k(n+1)} \frac{1}{j}</math>

This can be estimated using the integral test for convergence, or more directly by comparing it to the [[#Riemann Sum|integral]] of <math>1/x</math> from <math>k</math> to <math>k(n+1)</math>:

:<math>\lim_{{k \to \infty}} \sum_{{j=k+1}}^{k(n+1)} \frac{1}{j} = \int_k^{k(n+1)} \frac{dx}{x} = \ln(k(n+1)) - \ln(k) = \ln\left(\frac{k(n+1)}{k}\right) = \ln(n+1)</math>

As the window's lower bound begins at <math>k+1</math> and the upper bound extends to <math>k(n+1)</math>, both of which tend toward infinity as <math>k \to \infty</math>, the summation window encompasses an increasingly vast portion of the smallest possible terms of the harmonic series (those with astronomically large denominators), creating a discrete sum that stretches towards infinity, which mirrors how continuous integrals accumulate value across an infinitesimally fine partitioning of the domain. In the limit, the interval is effectively from <math>1</math> to <math>n+1</math> where the onset <math>k</math> implies this minimally discrete region.