Is there an approximation in terms of nth terms at the harmonic series (and simple to prove)?

Question

$$H(n)=\frac 11+\frac 12+\frac 13+...+\frac 1n$$ I know some approximation to $H(n)$ like below $$\ln(n)=\int_{1}^n \frac{1}{x}dx < H_n< 1+\int_{1}^n \frac{1}{x}dx=1+\ln(n).$$ or $$H_n=\gamma +\log \left({n}\right)+\frac{1}{2 n}+O\left(\frac{1}{n^2}\right)$$ but I don't know-how is the proof.

or is there a better approximation for $H(n)$ .
I mean better as 'simple to describe'.

Thanks in advance for any proof,reference,Idea or link...

Answer 1

You already proved that $H_n=\ln n+\mathcal{O}(1)$. Moreover $$ \forall n\in\mathbb{N}^*,\,\frac{1}{n+1}\leqslant\int_n^{n+1}\frac{dx}{x}\leqslant\frac{1}{n} $$ Thus, if $u_n=\frac{1}{n}-\int_n^{n+1}\frac{dx}{x}$ we have $$ 0\leqslant u_n\leqslant \frac{1}{n}-\frac{1}{n+1} $$ and the series $\sum u_n$ converges, we define $\gamma:=\sum_{n=1}^{+\infty}u_n$. However $$ \sum_{k=1}^{n}u_k=H_n-\ln(n+1) $$ so that $H_n=\ln n+\gamma+o(1)$. Moreover, if $v_n=u_n-\ln n$ we have $v_{n+1}-v_n\underset{n\rightarrow +\infty}{\sim}\frac{1}{2n^2}$ so that $$ \sum_{k=n}^{+\infty}{(v_{k+1}-v_k)}\underset{n\rightarrow +\infty}{\sim}\sum_{k=n}^{+\infty}\frac{1}{2k^2}\underset{n\rightarrow +\infty}{\sim}\frac{1}{2n} $$ We finally have $H_n=\ln n+\gamma+\frac{1}{2n}+o\left(\frac{1}{n}\right)$ One can show that $$ H_n \underset{n\rightarrow +\infty}{\sim} \ln n+\gamma+\frac{1}{2n}-\sum_{k=1}^{+\infty}\frac{B_{2k}}{2kn^{2k}} $$ using Euler-Maclaurin formula :

https://en.wikipedia.org/wiki/Harmonic_number https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula

Answer 2

The method I like is to prove (not hard) that $H_n - \log n$ begins high and decreases, while $H_n - \log (n+1) $ begins low and increases. The result is that both approach a a constant, that we call gamma. As we also have $$ H_n - \log (n+1) \; \; < \; \; H_n - \log n \; \; , \; \; $$ the decreasing/increasing behavior tells us that we can prove, for any integers $m,n \geq 1,$ that $$ H_m - \log (m+1) \; \; < \; \; H_n - \log n \; \; , \; \; $$ since for some $k $ bigger than both $m,n$ we get $$ H_m - \log (m+1) \; \; < \; \; H_k - \log (k+1) \; \; < \; \; H_k - \log k \; \; < \; \; H_n - \log n \; \; . \; \; $$

jagy@phobeusjunior:~$ ./gummy_bears
1    Hn - log n :   1                       Hn - log(n+1) :     0.3068528194400547
2    Hn - log n :   0.8068528194400547      Hn - log(n+1) :     0.4013877113318903
3    Hn - log n :   0.7347210446652236      Hn - log(n+1) :     0.4470389722134426
4    Hn - log n :   0.6970389722134425      Hn - log(n+1) :     0.4738954208992326
5    Hn - log n :   0.6738954208992328      Hn - log(n+1) :     0.4915738641052782
6    Hn - log n :   0.6582405307719448      Hn - log(n+1) :     0.5040898509446864
7    Hn - log n :   0.6469469938018292      Hn - log(n+1) :     0.5134156011773066
8    Hn - log n :   0.6384156011773066      Hn - log(n+1) :     0.5206325655209232
9    Hn - log n :   0.6317436766320343      Hn - log(n+1) :     0.526383160974208
10   Hn - log n :   0.6263831609742081      Hn - log(n+1) :     0.5310729811698832
11   Hn - log n :   0.621982072078974       Hn - log(n+1) :     0.5349706950893443
12   Hn - log n :   0.6183040284226777      Hn - log(n+1) :     0.5382613207491413
13   Hn - log n :   0.6151843976722184      Hn - log(n+1) :     0.5410764255184966
14   Hn - log n :   0.6125049969470682      Hn - log(n+1) :     0.5435121254601167
15   Hn - log n :   0.6101787921267836      Hn - log(n+1) :     0.5456402709892124
16   Hn - log n :   0.6081402709892124      Hn - log(n+1) :     0.5475156491727776
17   Hn - log n :   0.6063391785845421      Hn - log(n+1) :     0.5491807647445934
18   Hn - log n :   0.6047363203001488      Hn - log(n+1) :     0.550669099029873
19   Hn - log n :   0.6033006779772416      Hn - log(n+1) :     0.5520073835896911
20   Hn - log n :   0.602007383589691       Hn - log(n+1) :     0.5532172194202589
21   Hn - log n :   0.6008362670393064      Hn - log(n+1) :     0.5543162514044135
22   Hn - log n :   0.599770796858959       Hn - log(n+1) :     0.5553190342881251
23   Hn - log n :   0.5987972951576905      Hn - log(n+1) :     0.5562376807388946
24   Hn - log n :   0.5979043474055611      Hn - log(n+1) :     0.5570823528853059
25   Hn - log n :   0.597082352885306       Hn - log(n+1) :     0.5578616397320247
26   Hn - log n :   0.5963231781935631      Hn - log(n+1) :     0.558582850210716
27   Hn - log n :   0.5956198872477532      Hn - log(n+1) :     0.5592522430768784
28   Hn - log n :   0.594966528791164       Hn - log(n+1) :     0.5598752089798938
29   Hn - log n :   0.5943579676005833      Hn - log(n+1) :     0.560456415924902
30   Hn - log n :   0.5937897492582352      Hn - log(n+1) :     0.5609999264352443
31   Hn - log n :   0.5932579909513738      Hn - log(n+1) :     0.5615092926367935
32   Hn - log n :   0.5927592926367935      Hn - log(n+1) :     0.5619876339700398
33   Hn - log n :   0.5922906642730701      Hn - log(n+1) :     0.562437701123389
34   Hn - log n :   0.5918494658292712      Hn - log(n+1) :     0.5628619289560189
35   Hn - log n :   0.5914333575274474      Hn - log(n+1) :     0.563262480560751
36   Hn - log n :   0.5910402583385287      Hn - log(n+1) :     0.5636412841504143
37   Hn - log n :   0.5906683111774415      Hn - log(n+1) :     0.5640000640952801
38   Hn - log n :   0.5903158535689642      Hn - log(n+1) :     0.5643403671657036
39   Hn - log n :   0.5899813928067291      Hn - log(n+1) :     0.5646635848224393
40   Hn - log n :   0.5896635848224396      Hn - log(n+1) :     0.564970972232068
41   Hn - log n :   0.5893612161345071      Hn - log(n+1) :     0.5652636645554466
42   Hn - log n :   0.5890731883649704      Hn - log(n+1) :     0.5655426909547763
43   Hn - log n :   0.5887985049082647      Hn - log(n+1) :     0.5658089866835659
44   Hn - log n :   0.5885362594108384      Hn - log(n+1) :     0.5660634035587798
45   Hn - log n :   0.588285625781002       Hn - log(n+1) :     0.5663067190622267
46   Hn - log n :   0.588045849497009       Hn - log(n+1) :     0.5665396442760454
47   Hn - log n :   0.587816240020726       Hn - log(n+1) :     0.5667628308228936
48   Hn - log n :   0.5875961641562266      Hn - log(n+1) :     0.566976876953491
49   Hn - log n :   0.5873850402187969      Hn - log(n+1) :     0.5671823329012774
50   Hn - log n :   0.587182332901277       Hn - log(n+1) :     0.5673797056050973
51   Hn - log n :   0.5869875487423521      Hn - log(n+1) :     0.5675694628852506
52   Hn - log n :   0.5868002321160197      Hn - log(n+1) :     0.5677520371453252
53   Hn - log n :   0.5866199616736273      Hn - log(n+1) :     0.5679278286614747
54   Hn - log n :   0.5864463471799929      Hn - log(n+1) :     0.5680972085117963
55   Hn - log n :   0.5862790266936149      Hn - log(n+1) :     0.5682605211909366
56   Hn - log n :   0.5861176640480799      Hn - log(n+1) :     0.5684180869486789
57   Hn - log n :   0.5859619465978013      Hn - log(n+1) :     0.5685702038859322
58   Hn - log n :   0.5858115831962774      Hn - log(n+1) :     0.5687171498369772
59   Hn - log n :   0.58566630237935        Hn - log(n+1) :     0.5688591840629689
60   Hn - log n :   0.5855258507296355      Hn - log(n+1) :     0.5689965487784249
61   Hn - log n :   0.5853899914013755      Hn - log(n+1) :     0.5691294705295952
62   Hn - log n :   0.5852585027876601      Hn - log(n+1) :     0.569258161441219
63   Hn - log n :   0.5851311773142348      Hn - log(n+1) :     0.5693828203460957
64   Hn - log n :   0.5850078203460957      Hn - log(n+1) :     0.5695036338101304
65   Hn - log n :   0.5848882491947457      Hn - log(n+1) :     0.5696207770639573