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Conditional Convergence


A series is said to be conditionally convergent iff it is convergent, the series of its positive terms diverges to positive infinity, and the series of its negative terms diverges to negative infinity.

Examples of conditionally convergent series include the alternating harmonic series

 sum_(n=1)^infty((-1)^(n+1))/n=ln2

and the logarithmic series

 sum_(n=1)^infty((-1)^nlnn)/n=gammaln2-1/2(ln2)^2,

where gamma is the Euler-Mascheroni constant.

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. The Riemann series theorem can be proved by first taking just enough positive terms to exceed the desired limit, then taking just enough negative terms to go below the desired limit, and iterating this procedure. Since the terms of the original series tend to zero, the rearranged series converges to the desired limit. A slight variation works to make the new series diverge to positive infinity or to negative infinity.


See also

Absolute Convergence, Alternating Series, Convergence Tests, Convergent Series, Divergent Series, Riemann Series Theorem, 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, 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.

Referenced on Wolfram|Alpha

Conditional Convergence

Cite this as:

Weisstein, Eric W. "Conditional Convergence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConditionalConvergence.html

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