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The series of numbers
 
The series of numbers
 
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\begin{equation}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046540/h0465401.png" /></td> </tr></table>
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\sum_{k=1}^{\infty}\frac{1}{k}.
 
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\end{equation}
 
Each term of the harmonic series (beginning with the second) is the [[Harmonic mean|harmonic mean]] of its two contiguous terms (hence the name harmonic series). The harmonic series is divergent (G. Leibniz, 1673), and its partial sums
 
Each term of the harmonic series (beginning with the second) is the [[Harmonic mean|harmonic mean]] of its two contiguous terms (hence the name harmonic series). The harmonic series is divergent (G. Leibniz, 1673), and its partial sums
 
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\begin{equation}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046540/h0465402.png" /></td> </tr></table>
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S_n = \sum_{k=1}^n\frac{1}{k}
 
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\end{equation}
increase as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046540/h0465403.png" /> (L. Euler, 1740). There exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046540/h0465404.png" />, known as the [[Euler constant|Euler constant]], such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046540/h0465405.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046540/h0465406.png" />. The series
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increase as $\ln n$ (L. Euler, 1740). There exists a constant $\gamma>0$, known as the [[Euler constant|Euler constant]], such that $S_n = \ln n + \gamma + \varepsilon_n$, where $\lim\limits_{n\to\infty}\varepsilon_n = 0$. The series
 
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\begin{equation}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046540/h0465407.png" /></td> </tr></table>
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\sum_{k=1}^{\infty}\frac{1}{k^{\alpha}}
 
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\end{equation}
is called the generalized harmonic series; it is convergent for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046540/h0465408.png" /> and divergent for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046540/h0465409.png" />.
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is called the generalized harmonic series; it is convergent for $\alpha>1$ and divergent for $\alpha\leq1$.
 
 
 
 
  
 
====Comments====
 
====Comments====
For a proof of the expression for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046540/h04654010.png" /> see, e.g., [[#References|[a1]]], Thm. 422. Note that the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046540/h04654011.png" /> extended over all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046540/h04654012.png" /> diverges also; see, e.g., [[#References|[a1]]], Thm. 427, for an expression of its partial sums.
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For a proof of the expression for $S_n$ see, e.g., [[#References|[a1]]], Thm. 422. Note that the series $\sum 1/p$ extended over all prime numbers $p$ diverges also; see, e.g., [[#References|[a1]]], Thm. 427, for an expression of its partial sums.
  
Generalized harmonic series are often used to test whether a given series is convergent or divergent by estimating in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046540/h04654013.png" /> the order of the terms of the given series; see [[Series|Series]].
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Generalized harmonic series are often used to test whether a given series is convergent or divergent by estimating in terms of $1/n^{\alpha}$ the order of the terms of the given series; see [[Series|Series]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)</TD></TR></table>

Latest revision as of 10:24, 10 December 2012


The series of numbers \begin{equation} \sum_{k=1}^{\infty}\frac{1}{k}. \end{equation} Each term of the harmonic series (beginning with the second) is the harmonic mean of its two contiguous terms (hence the name harmonic series). The harmonic series is divergent (G. Leibniz, 1673), and its partial sums \begin{equation} S_n = \sum_{k=1}^n\frac{1}{k} \end{equation} increase as $\ln n$ (L. Euler, 1740). There exists a constant $\gamma>0$, known as the Euler constant, such that $S_n = \ln n + \gamma + \varepsilon_n$, where $\lim\limits_{n\to\infty}\varepsilon_n = 0$. The series \begin{equation} \sum_{k=1}^{\infty}\frac{1}{k^{\alpha}} \end{equation} is called the generalized harmonic series; it is convergent for $\alpha>1$ and divergent for $\alpha\leq1$.

Comments

For a proof of the expression for $S_n$ see, e.g., [a1], Thm. 422. Note that the series $\sum 1/p$ extended over all prime numbers $p$ diverges also; see, e.g., [a1], Thm. 427, for an expression of its partial sums.

Generalized harmonic series are often used to test whether a given series is convergent or divergent by estimating in terms of $1/n^{\alpha}$ the order of the terms of the given series; see Series.

References

[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979)
How to Cite This Entry:
Harmonic series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_series&oldid=17912
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article