Calculus How To

Alternating Harmonic Series

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The alternating harmonic series is the sum:

alternating harmonic series

Which converges (i.e. settle on a certain number) to ln(2). It is the x = 1 case of the Mercator series, and also a special case of the Dirichlet eta function.

The image below shows the first fourteen partial sums of this series. Ln(2) is shown in red. The more terms of the sequence are added up, the closer we get to the line ln(2). With a large number of terms, the difference is indistinguishable.

alternating harmonic series

The harmonic series and alternating harmonic series both get their names from the harmonic wavelengths of music, which follow the same pattern.

Conditional Convergence of the Alternating Harmonic Series

It’s important to note that although the alternating harmonic series does converge to ln 2, it only converges conditionally. It is not absolutely convergent, for it is possible to rearrange the terms of the series so that we can come up with any answer whatsoever.

Assume, for example, you wanted it to converge to 2.0. Then you could sum a few positive terms, to get something slightly greater than 2. Add one negative term, so that the sum is just below 2; then add enough positive terms to make it go above 2 again. Continue doing this, going closer and closer to 2, and if you repeat forever—as you can, since you have an infinite number of terms—it will converge to 2.
conditional convergence


References

Hudleson, Matt. Proof Without Words. Mathematics Magazine 83 (2010) 294. doi:10.4169/002557010X521831. Mathematical Association of America. Retrieved from https://www.maa.org/sites/default/files/Hudleson-MMz-201007804.pdf on August 21, 2019
Harrison. Lecture 2, Sequences and Series. Retrieved from https://www.ch.ic.ac.uk/harrison/Teaching/L2_Seq-Series-2.pdf on August 21, 2019

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Stephanie Glen. "Alternating Harmonic Series" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/alternating-harmonic-series/
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