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[parent] sum of series depends on order (Example)

According to the Leibniz' test, the alternating series

$\displaystyle 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7} -\frac{1}{8}+\frac{1}{9}-\frac{1}{10}+\frac{1}{11}-\frac{1}{12}+-\cdots$
is convergent and has a positive sum ($ = \ln{2}$; see the natural logarithm). Denote it by $ S$. We can group pairwise its terms and multiply each term by $ \frac{1}{2}$ getting the two series $ S = (1-\frac{1}{2})+(\frac{1}{3}-\frac{1}{4})+(\frac{1}{5}-\frac{1}{6})+(\frac{1}{7} -\frac{1}{8})+(\frac{1}{9}-\frac{1}{10})+\cdots,$

$ \frac{1}{2}S = \frac{1}{2}-\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\frac{1}{10}-+\cdots.$

Then we add these two series termwise getting the sum

$ 1\frac{1}{2}S = 1+\frac{1}{3}-\frac{2}{4}+\frac{1}{5}+\frac{1}{7} -\frac{2}{8}+\frac{1}{9}+\frac{1}{11}-\frac{2}{12}+\cdots.$

Hence, this last series contains exactly the same terms as the original, but its sum is fifty percent greater. This is possible because the original series is not absolutely convergent: the series which is formed of the absolute values of its terms is the divergent harmonic series.

P. S. - For justification of the used manipulations of the series, see the parent entry.



"sum of series depends on order" is owned by pahio.
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See Also: convergent series, order of factors in infinite product, alternating harmonic series

Keywords:  conditional convergence

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Cross-references: harmonic series, divergent, absolute values, absolutely convergent, percent, series, sum, positive, convergent, alternating series
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This is version 12 of sum of series depends on order, born on 2004-11-23, modified 2008-03-08.
Object id is 6520, canonical name is SumOfSeriesDependsOnOrder.
Accessed 2978 times total.

Classification:
AMS MSC: 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
 26A06 (Real functions :: Functions of one variable :: One-variable calculus)
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