TOPICS
Search

Divergent Series

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

A divergent series is a series which is not convergent. Series may diverge by marching off to infinity or by oscillating. Divergent series have some curious properties. For example, formally grouping the terms of 1-1+1-1+1-... gives both (1-1)+(1-1)+(1-1)+...=0 and 1-(1-1)-(1-1)+...=1.

The Riemann series theorem states that, by a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge.

No less an authority than N. H. Abel wrote "The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever" (Gardner 1984, p. 171; Hoffman 1998, p. 218). However, divergent series can actually be "summed" rigorously by using extensions to the usual summation rules (e.g., Abel-regularized sums and Cesàro sums). For example, the divergent series 1-1+1-1+1-... has both an Abel-regularized sum and a Cesàro sum of 1/2.

See also

Abel-Regularized Sum, Absolute Convergence, Cesàro Sum, Conditional Convergence, Convergent Series, Divergent Sequence, Logarithmic Series, Regularized Sum

Explore with Wolfram|Alpha

References

Boos, J. Classical and Modern Methods in Summability. Assisted by P. Cass. Oxford, England: Oxford University Press, 2000. https://doi.org/10.1093/oso/9780198501657.001.0001.Bromwich, T. J. I'A. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, 1991.Ford, W. B. Studies on Divergent Series and Summability and the Asymptotic Developments of Functions Defined by Maclaurin Series. New York: Chelsea, 1960.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 170-171, 1984.Hardy, G. H. Divergent Series. New York: Oxford University Press, 1949.Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, 1998.Tucciarone, J. "The Development of the Theory of Summable Divergent Series from 1880 to 1925." Arch. Hist. Exact Sci. 10, 1-40, 1973. https://doi.org/10.1007/BF00343405.

Referenced on Wolfram|Alpha

Divergent Series

Cite this as:

Weisstein, Eric W. "Divergent Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DivergentSeries.html

Subject classifications