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Absolute Convergence


A series sum_(n)u_n is said to converge absolutely if the series sum_(n)|u_n| converges, where |u_n| denotes the absolute value. If a series is absolutely convergent, then the sum is independent of the order in which terms are summed. Furthermore, if the series is multiplied by another absolutely convergent series, the product series will also converge absolutely.


See also

Conditional Convergence, Convergent Series, Riemann Series Theorem

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References

Bromwich, T. J. I'A. and MacRobert, T. M. "Absolute Convergence." Ch. 4 in An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, pp. 69-77, 1991.Jeffreys, H. and Jeffreys, B. S. "Absolute Convergence." §1.051 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 16, 1988.

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Absolute Convergence

Cite this as:

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

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