By a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired
value, or to diverge.
For example,
converges to ,
but the same series can be rearranged to
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.Referenced on Wolfram|Alpha
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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