Editing Harmonic series (mathematics) (section)

==Partial sums==
{{main|Harmonic number}}
{| class="wikitable" style="margin:0 0 0 1em; text-align:right; float:right;"
! rowspan="2" style="padding-top:1em;"|<math>n</math> !! colspan="4"|Partial sum of the harmonic series, <math>H_n</math>
|-
! colspan="2"|expressed as a fraction !! decimal !! relative size
|-
| style="text-align:right;"|1 || style="text-align:center;" colspan="2"|1 || {{0|~}}{{bartable|1||20}}
|-
| style="text-align:right;"|2 || style="border-right:none;text-align:right;padding-right:0;"|3 || style="border-left:none;text-align:left;padding-left:0;"|/2 || {{bartable|1.5||20}}
|-
| style="text-align:right;"|3 || style="border-right:none;text-align:right;padding-right:0;"|11 || style="border-left:none;text-align:left;padding-left:0;"|/6 || ~{{bartable|1.83333||20}}
|-
| style="text-align:right;"|4 || style="border-right:none;text-align:right;padding-right:0;"|25 || style="border-left:none;text-align:left;padding-left:0;"|/12 || ~{{bartable|2.08333||20}}
|-
| style="text-align:right;"|5 || style="border-right:none;text-align:right;padding-right:0;"|137 || style="border-left:none;text-align:left;padding-left:0;"|/60 || ~{{bartable|2.28333||20}}
|-
| style="text-align:right;"|6 || style="border-right:none;text-align:right;padding-right:0;"|49 || style="border-left:none;text-align:left;padding-left:0;"|/20 || {{bartable|2.45||20}}
|-
| style="text-align:right;"|7 || style="border-right:none;text-align:right;padding-right:0;"|363 || style="border-left:none;text-align:left;padding-left:0;"|/140 || ~{{bartable|2.59286||20}}
|-
| style="text-align:right;"|8 || style="border-right:none;text-align:right;padding-right:0;"|761 || style="border-left:none;text-align:left;padding-left:0;"|/280 || ~{{bartable|2.71786||20}}
|-
| style="text-align:right;"|9 || style="border-right:none;text-align:right;padding-right:0;"|7129 || style="border-left:none;text-align:left;padding-left:0;"|/2520 || ~{{bartable|2.82897||20}}
|-
| style="text-align:right;"|10 || style="border-right:none;text-align:right;padding-right:0;"|7381 || style="border-left:none;text-align:left;padding-left:0;"|/2520 || ~{{bartable|2.92897||20}}
|-
| style="text-align:right;"|11 || style="border-right:none;text-align:right;padding-right:0;"|83711 || style="border-left:none;text-align:left;padding-left:0;"|/27720 || ~{{bartable|3.01988||20}}
|-
| style="text-align:right;"|12 || style="border-right:none;text-align:right;padding-right:0;"|86021 || style="border-left:none;text-align:left;padding-left:0;"|/27720 || ~{{bartable|3.10321||20}}
|-
| style="text-align:right;"|13 || style="border-right:none;text-align:right;padding-right:0;"|1145993 || style="border-left:none;text-align:left;padding-left:0;"|/360360 || ~{{bartable|3.18013||20}}
|-
| style="text-align:right;"|14 || style="border-right:none;text-align:right;padding-right:0;"|1171733 || style="border-left:none;text-align:left;padding-left:0;"|/360360 || ~{{bartable|3.25156||20}}
|-
| style="text-align:right;"|15 || style="border-right:none;text-align:right;padding-right:0;"|1195757 || style="border-left:none;text-align:left;padding-left:0;"|/360360 || ~{{bartable|3.31823||20}}
|-
| style="text-align:right;"|16 || style="border-right:none;text-align:right;padding-right:0;"|2436559 || style="border-left:none;text-align:left;padding-left:0;"|/720720 || ~{{bartable|3.38073||20}}
|-
| style="text-align:right;"|17 || style="border-right:none;text-align:right;padding-right:0;"|42142223 || style="border-left:none;text-align:left;padding-left:0;"|/12252240 || ~{{bartable|3.43955||20}}
|-
| style="text-align:right;"|18 || style="border-right:none;text-align:right;padding-right:0;"|14274301 || style="border-left:none;text-align:left;padding-left:0;"|/4084080 || ~{{bartable|3.49511||20}}
|-
| style="text-align:right;"|19 || style="border-right:none;text-align:right;padding-right:0;"|275295799 || style="border-left:none;text-align:left;padding-left:0;"|/77597520 || ~{{bartable|3.54774||20}}
|-
| style="text-align:right;"|20 || style="border-right:none;text-align:right;padding-right:0;"|55835135 || style="border-left:none;text-align:left;padding-left:0;"|/15519504 || ~{{bartable|3.59774||20}}
|}
<!-- Python script to generate n from 2 to 30:
import fractions
numerator = 1; denominator = 1
for i in range(2, 30 + 1):
 numerator = numerator * i + denominator; denominator *= i; gcd = fractions.gcd(numerator, denominator); numerator /= gcd; denominator /= gcd
 decimal   = ('{}' if (i < 3 or i == 6) else '{:.5f}').format(float(numerator) / denominator); exact = '' if (i < 3 or i == 6) else '~'
 print('|-\n| style="text-align:right;"|{} || style="border-right:none;text-align:right;padding-right:0;"|{:,} || style="border-left:none;text-align:left;padding-left:0;"|/{:,} || {}{{{{bartable|{}||20}}}}'.
       format(i, numerator, denominator, exact, decimal).replace(',', ''))
-->
Adding the first <math>n</math> terms of the harmonic series produces a [[partial sum]], called a [[harmonic number]] and {{nowrap|denoted <math>H_n</math>:{{r|knuth}}}}
<math display=block>H_n = \sum_{k = 1}^n \frac{1}{k}.</math>

===Growth rate===
These numbers grow very slowly, with [[logarithmic growth]], as can be seen from the integral test.{{r|bressoud}} More precisely, by the [[Euler–Maclaurin formula]],
<math display=block>H_n = \ln n + \gamma + \frac{1}{2n} - \varepsilon_n</math>
where <math>\gamma\approx 0.5772</math> is the [[Euler–Mascheroni constant]] and <math>0\le\varepsilon_n\le 1/(8n^2)</math> which approaches 0 as <math>n</math> goes to infinity.{{r|boawre}}

===Divisibility===
No harmonic numbers are integers except for {{nowrap|<math>H_1=1</math>.{{r|havil|osler}}}} One way to prove that <math>H_n</math> is not an integer is to consider the highest [[power of two]] <math>2^k</math> in the range from {{nowrap|1 to <math>n</math>.}} If <math>M</math> is the [[least common multiple]] of the numbers from {{nowrap|1 to <math>n</math>,}} then
<math>H_k</math> can be rewritten as a sum of fractions with equal denominators <math display=block>H_n=\sum_{i=1}^n \tfrac{M/i}{M}</math>
in which only one of the numerators, {{nowrap|<math>M/2^k</math>,}} is odd and the rest are even, and {{nowrap|(when <math>n>1</math>)}} <math>M</math> is itself even. Therefore, the result is a fraction with an odd numerator and an even denominator, which cannot be an integer.{{r|havil}} More generally, any sequence of consecutive integers has a unique member divisible by a greater power of two than all the other sequence members, from which it follows by the same argument that no two harmonic numbers differ by an integer.{{r|osler}}

Another proof that the harmonic numbers are not integers observes that the denominator of <math>H_n</math> must be divisible by 
all [[prime number]]s greater than <math>n/2</math> and less than or equal to <math>n</math>, and uses [[Bertrand's postulate]] to prove that this set of primes is non-empty. The same argument implies more strongly that, except for <math>H_1=1</math>, <math>H_2=1.5</math>, and <math>H_6=2.45</math>, no harmonic number can have a [[terminating decimal]] representation.{{r|havil}} It has been conjectured that every prime number divides the numerators of only a finite subset of the harmonic numbers, but this remains unproven.{{r|sanna}}

===Interpolation===
[[File:Psi0.png|thumb|upright|The [[digamma function]] on the complex numbers]]
The [[digamma function]] is defined as the [[logarithmic derivative]] of the [[gamma function]]
<math display=block>\psi(x)=\frac{d}{dx}\ln\big(\Gamma(x)\big)=\frac{\Gamma'(x)}{\Gamma(x)}.</math>
Just as the gamma function provides a continuous [[interpolation]] of the [[factorial]]s, the digamma function provides a continuous interpolation of the harmonic numbers, in the sense that {{nowrap|<math>\psi(n)=H_{n-1}-\gamma</math>.{{r|ross}}}}
This equation can be used to extend the definition to harmonic numbers with rational indices.<ref>{{cite journal|first1=Anthony|last1=Sofo|first2=H. M. |last2=Srivastava|title=A family of shifted harmonic sums|year=2015|journal=The Ramanujan Journal|volume=37|pages=89–108|doi=10.1007/s11139-014-9600-9|s2cid=254990799 }}</ref>