Editing Harmonic number (section)

==Calculation==
An integral representation given by [[Euler]]<ref>{{citation|title=How Euler Did It|series=MAA Spectrum|first=C. Edward|last=Sandifer|publisher=Mathematical Association of America|year=2007|isbn=9780883855638|page=206|url=https://books.google.com/books?id=sohHs7ExOsYC&pg=PA206}}.</ref> is
<math display="block"> H_n = \int_0^1 \frac{1 - x^n}{1 - x}\,dx. </math>

The equality above is straightforward by the simple [[algebraic identity]]
<math display="block"> \frac{1-x^n}{1-x}=1+x+\cdots +x^{n-1}.</math>

Using the substitution {{math|1=''x'' = 1 &minus; ''u''}}, another expression for {{math|''H''<sub>''n''</sub>}} is
<math display="block">\begin{align}
H_n &= \int_0^1 \frac{1 - x^n}{1 - x}\,dx = \int_0^1\frac{1-(1-u)^n}{u}\,du \\[6pt]
&= \int_0^1\left[\sum_{k=1}^n \binom nk (-u)^{k-1}\right]\,du 
= \sum_{k=1}^n \binom nk \int_0^1 (-u)^{k-1}\,du \\[6pt]
&= \sum_{k=1}^n \binom nk \frac{(-1)^{k-1}}{k}.
\end{align}
</math>

[[File:Integral Test.svg|thumb|Graph demonstrating a connection between harmonic numbers and the [[natural logarithm]]. The harmonic number {{math|''H''<sub>''n''</sub>}} can be interpreted as a [[Riemann sum]] of the integral: <math>\int_1^{n+1} \frac{dx}{x} = \ln(n+1).</math>]]
The {{mvar|n}}th harmonic number is about as large as the [[natural logarithm]] of {{mvar|n}}. The reason is that the sum is approximated by the [[integral]]
<math display="block">\int_1^n \frac{1}{x}\, dx,</math>
whose value is {{math|ln ''n''}}.

The values of the sequence {{math|''H''<sub>''n''</sub> − ln ''n''}} decrease monotonically towards the [[Limit of a sequence|limit]]
<math display="block"> \lim_{n \to \infty} \left(H_n - \ln n\right) = \gamma,</math>
where {{math|''γ'' ≈ 0.5772156649}} is the [[Euler–Mascheroni constant]]. The corresponding [[asymptotic expansion]] is
<math display="block">\begin{align}
   H_n &\sim \ln{n}+\gamma+\frac{1}{2n}-\sum_{k=1}^\infty \frac{B_{2k}}{2k n^{2k}}\\
   &=\ln{n}+\gamma+\frac{1}{2n}-\frac{1}{12n^2}+\frac{1}{120n^4}-\cdots,
 \end{align}</math>
where {{math|''B''<sub>''k''</sub>}} are the [[Bernoulli numbers]].

{{Reflist|group=note}}