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Divergent Series

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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, rearranging 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., so-called Abel and Cesàro sums). For example, the divergent series 1-1+1-1+1-... has both Abel and Cesàro sums of 1/2.

See also

Absolute Convergence, Conditional Convergence, Convergent Series, Divergent Sequence, Logarithmic Series, Regularization, Regularized Sum

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References

Bromwich, T. J. I'A. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, 1991.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.

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Divergent Series

Cite this as:

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

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