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Riemann Series Theorem

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By a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge.

For example,

S=1-1/2+1/3-1/4+1/5+...
(1)
=sum_(k=1)^(infty)((-1)^(k+1))/k
(2)
=ln2,
(3)

converges to ln2, but the same series can be rearranged to

S^'=(1-1/2-1/4)+(1/3-1/6-1/8)+(1/5-1/(10)-1/(12))+...
(4)
=sum_(k=1)^(infty)(1/(2k-1)-1/(4k-2)-1/(4k))
(5)
=sum_(k=1)^(infty)1/(4k(2k-1))
(6)
=1/2ln2,
(7)

so the series now converges to half of itself.

See also

Conditional Convergence, Divergent Series

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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, p. 74, 1991.Gardner, M. Martin Gardner's Sixth Book of Mathematical Games from Scientific American. New York: Scribner's, p. 171, 1971.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 102, 2003.

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Riemann Series Theorem

Cite this as:

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

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