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harmonic series (Topic)

The harmonic series

$\displaystyle \sum_{k=1}^\infty\frac{1}{k} \;=\; 1+\frac{1}{2}+\frac{1}{3}+\ldots$
satisfies the necessary condition of convergence

$\displaystyle \lim_{k\to\infty}a_n \;=\; 0$
for the series   $a_1+a_2+a_3+\ldots$ of real or complex terms:

$\displaystyle \lim_{k\to\infty}\frac{1}{k} \;=\; 0$
Nevertheless, the harmonic series diverges.  It is seen if we first group the terms with parentheses:

$\displaystyle 1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{4}\right) +\left(\frac{1... ...{1}{8}\right) +\left(\frac{1}{9}+\frac{1}{10}+\ldots+\frac{1}{16}\right)+\ldots$
Here, each parenthetic sum contains a number of terms twice as many as the preceding one.  The sum in the first parentheses is greater than  $2\cdot\frac{1}{4} = \frac{1}{2}$,  the sum in the second parentheses is greater than  $4\cdot\frac{1}{8} = \frac{1}{2}$;  thus one sees that the sum in all parentheses is greater than $\frac{1}{2}$.  Consequently, the partial sum of $n$ first terms exceeds any given real number, when $n$ is sufficiently big.

The divergence of the harmonic series is very slow, though.  Its speed may be illustrated by considering the difference

$\displaystyle \sum_{k=1}^{n-1}\frac{1}{k}-\!\int_1^n\frac{dx}{x} \;=\; \sum_{k=1}^{n-1}\frac{1}{k}-\ln{n}$
(see the diagram).  We know that $\ln{n}$ increases very slowly as $n \to \infty$ (e.g. $\ln{1\,000\,000\,000} \,\approx\, 20.7$).  The increasing of the partial sum $\sum_{k=1}^{n-1}\frac{1}{k}$ is about the same, since the limit

$\displaystyle \lim_{n\to\infty}\left(\sum_{k=1}^{n-1}\frac{1}{k}-\ln{n}\right)\;=\;\gamma$
is a little positive number

$\displaystyle \gamma \;=\; 0.5772156649...$
which is called the Euler constant or Euler–Mascheroni constant.

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"harmonic series" is owned by pahio.

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See Also: time-dependent harmonic oscillators

Also defines:  necessary condition of convergence, Euler constant

Attachments:
harmonic series diagram (Data Structure) by bci1

Cross-references: group
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This is version 6 of harmonic series, born on 2009-05-28, modified 2009-05-29.
Object id is 784, canonical name is HarmonicSeries.
Accessed 715 times total.

Classification:
Physics Classification: 02.30.-f (Function theory, analysis)
Pending Errata and Addenda
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