# alternating harmonic series

The alternating harmonic series is given by the following infinite series:

 $\sum_{i=1}^{\infty}\frac{(-1)^{n+1}}{n}.$ (1)

The series converges to $\ln 2$ and it is the prototypical example of a conditionally convergent series.

First, notice that the series is not absolutely convergent. By taking the absolute value of each term, we get the harmonic series, which is divergent. There are several ways to show this, and we invite the reader to the entry on harmonic series for further exploration.

Next, to show that the series (1) converges, we use the alternating series test (http://planetmath.org/AlternatingSeriesTest): since

 $\lim_{n\to\infty}\frac{1}{n}=0,$

the alternating series (1) converges.

Remarks.

• Other examples of conditionally convergent series can be discovered using variants of the alternating harmonic series. For instance, the following series

 $\sum_{i=1}^{\infty}\frac{(-1)^{n}n^{2}+\cos n}{n^{3}-n^{2}+e^{-n}}.$

can easily be shown to be conditionally convergent. Here is another example, more of a generalization, called the :

 $\sum_{i=1}^{\infty}\frac{(-1)^{n+1}}{n^{p}},$ (2)

where $p$ is non-negative real number. The convergence of the is tabulated below:

$p$ convergence
$(1,\infty)$ absolutely convergent
$(0,1]$ conditionally convergent
$0$ divergent

${}\end{center}\itemize@item UsingRiemannseriestheorem,% oneeasilyseesthatnoteveryconditionallyconvergentseriesisalternating.% Byappropriatelyrearrangingthealternatingharmonicseries,% onegetsaconditionallyconvergentseriesthatisnotalternating:define$σ:N→N${{{{{asfollows:\begin{displaymath}\sigma(i):=\left\{\begin{array}[]{ll}% \displaystyle{\frac{2i+1}{3}}&\textrm{if }2i\equiv-1\pmod{3}\\ \displaystyle{\frac{4i-2}{3}}&\textrm{if }2i\equiv 1\pmod{3}\\ \displaystyle{\frac{4i}{3}}&\textrm{otherwise}\\ }\@unrecurse}\right.\end{displaymath}Itcanbeshownthat\sigmaisabijection.Now,% leta_{n}bethenthtermofalternatingharmonicseries.Thenitisnothardtoseethat% \sum_{i=1}^{n}a_{\sigma(i)}\longrightarrow\frac{\ln 2}{2}\qquad\mbox{ as }% \qquad n\longrightarrow\infty.Thus,% itisconditionallyconvergentandyetitisnotalternating(thefirstthreetermsare1,\,% -\frac{1}{2},\,-\frac{1}{4}).\end{itemize}\begin{flushright}\begin{tabular}[]% {|ll|}\hline Title&alternating harmonic series\\ Canonical name&AlternatingHarmonicSeries\\ Date of creation&2013-03-22 17:53:25\\ Last modified on&2013-03-22 17:53:25\\ Owner&CWoo (3771)\\ Last modified by&CWoo (3771)\\ Numerical id&13\\ Author&CWoo (3771)\\ Entry type&Example\\ Classification&msc 26A06\\ Classification&msc 40A05\\ Synonym&alternating p-series\\ Related topic&AbsoluteConvergence\\ Related topic&MultiplicationOfSeries\\ Related topic&SumOfSeriesDependsOnOrder\\ Defines&alternating p-series\\ \hline}\end{tabular}}}\end{flushright}\end{document}\end{array}$
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