# alternating harmonic 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.

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

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

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