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

For example,

			(1)
			(2)
			(3)

converges to , but the same series can be rearranged to

			(4)
			(5)
			(6)
			(7)
			(8)

so the series now converges to half of itself.

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.

CITE THIS AS:

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