Why does $(\sum_{i=1}^x \frac{1}{i}) - \ln(x+\frac{1}{2})$ converge to $\gamma$ a lot faster than $(\sum_{i=1}^x \frac{1}{i}) - \ln(x)$

Question

$(\sum_{i=1}^x \frac{1}{i}) -\ln(x)$
10: 0,62638316097421
100: 0,58220733165153
1000: 0,57771558156821

$(\sum_{i=1}^x \frac{1}{i}) - \ln(x+\frac{1}{2})$
10: 0,57759299680478
100: 0,57721979014049
1000: 0,57721570652656

Answer 1

Let $H_n=\sum_{k=1}^n\frac1 k$. The Euler-Maclaurin summation formula gives $$H_n=\ln n+\gamma+\frac{1}{2n}+O(n^{-2}).$$ But $$\ln(n+1/2)=\ln n+\ln\left(1+\frac1{2n}\right)=\ln n+\frac1{2n}+O(n^{-2}).$$ Therefore $$H_n=\ln\left(n+\frac12\right)+\gamma+O(n^{-2}).$$ The error in the approximation is now $O(1/n^2)$ rather than $O(1/n)$.

Answer 2

This is not especially a number theory phenomenon. You are comparing a sum and an integral. Similar to this picture from a different question:

Now, before I paste in the diagram, just note that $f(n)$ is estimated by $\int_{n-1}^n f(t) dt \;,$ or by $\int_{n}^{n + 1} f(t) dt \;,$ but is estimated much better by $$\int_{n - \frac{1}{2}}^{n + \frac{1}{2} } \; f(t) \; dt \;.$$

Also note that the advantage of using the halves happens only in that first term. After that, both asymptotic expansions are similar, just even exponents. Here we go, $$ H_n = \gamma + \log \left( n + \frac{1}{2} \right) +\frac{1}{24\left( n + \frac{1}{2} \right)^2} - \frac{7}{960\left( n + \frac{1}{2} \right)^4} + O \left(\frac{1}{\left( n + \frac{1}{2} \right)^6} \right) $$ Note $$ \gamma \approx ‎0.57721566490153 $$

====================================================

  double h = 0.0;


  for(int n = 1; n <= 105; ++n)
  {
    h += 1.0 / n;
    double half = n + (1.0 / 2);
    double below = h - log(half) - 1 / ( 24 * half * half   );
    double above = below + 7 / ( 960 * half * half * half * half   );
    cout.precision(16);
    cout << setw(3) << n <<  setw(20) << h <<  setw(20) << below <<  setw(20) << above << endl;
  } 

=======================================================

  n       H_n               below gamma        above gamma  
  1                   1  0.5760163733733171  0.5774567025914241
  2                 1.5  0.5770426014591783   0.577229268125845
  3   1.833333333333333  0.5771690042937476   0.577217595158665
  4   2.083333333333333  0.5771983233883347  0.5772161052305335
  5   2.283333333333333  0.5772078306265884   0.577215799116901
  6                2.45  0.5772116298045227  0.5772157146288168
  7   2.592857142857143   0.577213381574137   0.577215686100886
  8   2.717857142857143   0.577214278092129  0.5772156749464131
  9   2.828968253968254  0.5772147748446763  0.5772156700700557
 10   2.928968253968254  0.5772150678554185  0.5772156677426398
 11   3.019877344877345  0.5772152496467665  0.5772156665501748
 12   3.103210678210678  0.5772153672357559  0.5772156659024226
 13   3.180133755133755  0.5772154460039576  0.5772156655328736
 14   3.251562326562327   0.577215500362116  0.5772156653130572
 15   3.318228993228994  0.5772155388494049  0.5772156651775194
 16   3.380728993228994  0.5772155667148677  0.5772156650912913
 17   3.439552522640758   0.577215587289521   0.577215665034905
 18   3.495108078196314  0.5772156027470478  0.5772156649971248
 19   3.547739657143682  0.5772156145413271  0.5772156649712566
 20   3.597739657143682  0.5772156236663827  0.5772156649531959
 21   3.645358704762729  0.5772156308153391  0.5772156649403632
 22   3.690813250217275  0.5772156364801514  0.5772156649310989
 23    3.73429151108684  0.5772156410156465  0.5772156649243132
 24   3.775958177753507  0.5772156446815149  0.5772156649192762
 25   3.815958177753507  0.5772156476703771  0.5772156649154918
 26   3.854419716215045  0.5772156501268676   0.577215664912616
 27   3.891456753252082  0.5772156521608239  0.5772156649104084
 28   3.927171038966368  0.5772156538565317  0.5772156649086969
 29   3.961653797587057  0.5772156552792992  0.5772156649073582
 30   3.994987130920391   0.577215656480186  0.5772156649063024
 31    4.02724519543652  0.5772156574994485  0.5772156649054635
 32    4.05849519543652  0.5772156583690728  0.5772156649047917
 33    4.08879822573955  0.5772156591146631    0.57721566490425
 34   4.118209990445433  0.5772156597568547  0.5772156649038104
 35   4.146781419016861   0.577215660312387  0.5772156649034516
 36   4.174559196794639  0.5772156607949213  0.5772156649031569
 37   4.201586223821666  0.5772156612156711   0.577215664902914
 38    4.22790201329535  0.5772156615838906   0.577215664902712
 39   4.253543038936376  0.5772156619072569  0.5772156649025438
 40   4.278543038936376  0.5772156621921698  0.5772156649024033
 41   4.302933282838815  0.5772156624439875  0.5772156649022848
 42   4.326742806648339  0.5772156626672178  0.5772156649021847
 43   4.349998620601827  0.5772156628656684  0.5772156649020997
 44     4.3727258933291  0.5772156630425677  0.5772156649020274
 45   4.394948115551322  0.5772156632006642  0.5772156649019654
 46   4.416687245986104  0.5772156633423058  0.5772156649019121
 47   4.437963841730785  0.5772156634695057  0.5772156649018663
 48   4.458797175064118  0.5772156635839953  0.5772156649018267
 49   4.479205338329423  0.5772156636872687  0.5772156649017924
 50   4.499205338329423  0.5772156637806188  0.5772156649017626
 51   4.518813181466678  0.5772156638651685  0.5772156649017368
 52   4.538043950697447  0.5772156639418947  0.5772156649017143
 53   4.556911875225749  0.5772156640116504  0.5772156649016947
 54   4.575430393744267  0.5772156640751811  0.5772156649016772
 55   4.593612211926086   0.577215664133143  0.5772156649016624
 56   4.611469354783229  0.5772156641861109  0.5772156649016494
 57   4.629013214432351  0.5772156642345918  0.5772156649016372
 58   4.646254593742697  0.5772156642790354  0.5772156649016271
 59    4.66320374628507  0.5772156643198376  0.5772156649016178
 60   4.679870412951736  0.5772156643573509  0.5772156649016096
 61   4.696263855574687  0.5772156643918882  0.5772156649016019
 62   4.712392887832752  0.5772156644237292  0.5772156649015958
 63   4.728265903705767  0.5772156644531214  0.5772156649015899
 64   4.743890903705767  0.5772156644802882  0.5772156649015847
 65   4.759275519090383  0.5772156645054282  0.5772156649015799
 66   4.774427034241898   0.577215664528721  0.5772156649015762
 67   4.789352407376227  0.5772156645503264  0.5772156649015726
 68   4.804058289729168  0.5772156645703888  0.5772156649015691
 69   4.818551043352357  0.5772156645890391  0.5772156649015661
 70   4.832836757638071  0.5772156646063945  0.5772156649015633
 71   4.846921264680325  0.5772156646225621  0.5772156649015612
 72   4.860810153569214  0.5772156646376377  0.5772156649015593
 73     4.8745087837062  0.5772156646517078  0.5772156649015567
 74   4.888022297219713  0.5772156646648526  0.5772156649015545
 75   4.901355630553047  0.5772156646771447  0.5772156649015531
 76   4.914513525289889  0.5772156646886492  0.5772156649015519
 77   4.927500538276902  0.5772156646994255  0.5772156649015505
 78   4.940321051097415  0.5772156647095285  0.5772156649015492
 79   4.952979278945516   0.577215664719008  0.5772156649015481
 80   4.965479278945517  0.5772156647279097  0.5772156649015472
 81   4.977824957957862  0.5772156647362753  0.5772156649015462
 82   4.990020079909081  0.5772156647441425  0.5772156649015449
 83   5.002068272680166  0.5772156647515477  0.5772156649015442
 84   5.013973034584928  0.5772156647585223  0.5772156649015433
 85   5.025737740467281   0.577215664765096  0.5772156649015425
 86   5.037365647444025  0.5772156647712965  0.5772156649015419
 87   5.048859900317588  0.5772156647771484   0.577215664901541
 88   5.060223536681224  0.5772156647826757  0.5772156649015406
 89   5.071459491737405  0.5772156647878995  0.5772156649015404
 90   5.082570602848516  0.5772156647928391  0.5772156649015399
 91   5.093559613837527  0.5772156647975133  0.5772156649015394
 92   5.104429179054918  0.5772156648019391  0.5772156649015392
 93   5.115181867226961  0.5772156648061318  0.5772156649015389
 94   5.125820165099301  0.5772156648101061  0.5772156649015384
 95   5.136346480888775  0.5772156648138752  0.5772156649015376
 96   5.146763147555442  0.5772156648174528  0.5772156649015376
 97   5.157072425905957  0.5772156648208494  0.5772156649015373
 98    5.16727650753861  0.5772156648240766  0.5772156649015374
 99   5.177377517639621   0.577215664827144  0.5772156649015374
100   5.187377517639621  0.5772156648300606   0.577215664901537
101   5.197278507738631  0.5772156648328358  0.5772156649015368
102   5.207082429307258  0.5772156648354776  0.5772156649015365
103   5.216791167171336  0.5772156648379941  0.5772156649015368
104   5.226406551786721  0.5772156648403919   0.577215664901537
105   5.235930361310531  0.5772156648426773  0.5772156649015369
  n       H_n               below gamma        above gamma  

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