Absolute Convergence

Pronunciation: /æb səˈlut kənˈvɜrdʒ əns/ ?

A series is said to converge absolutely if the sum of the absolute value of the terms is convergent.[1] In algebraic notation: A series The sum from n=0 to infinity of a sub n. is said to converge absolutely if The sum from n=0 to infinity of the absolute value of a sub n is less than infinity..

A series that converges absolutely, also converges itself. If two absolutely convergent series are multiplied together, the resulting series is also absolutely convergent.

Example

Start with the series The sum from n=0 to infinity of (-1)^n/(3^n).. If the series The sum from n=0 to infinity of the absolute value of (-1)^n/(3^n). converges, then the series is absolutely convergent. Since The sum from n=0 to infinity of the absolute value of (-1)^n/(3^n)=1+1/3+1/9+1/27+.... is a infinite geometric series with a ratio of 1/3, it converges to 1/(1-1/3)=3/2, So The sum from n=0 to infinity of (-1)^n/(3^n). converges absolutely.

References

  1. absolute convergence. merriam-webster.com. Encyclopedia Britannica. (Accessed: 2010-12-07). http://www.merriam-webster.com/dictionary/absolute%20convergence.
  2. Manning, Henry Parker, Ph.D.. Irrational Numbers and Their Representation by Series and Sequences, pg 89. John Wiley & Sons, 1906. (Accessed: 2009-12-24). http://www.archive.org/stream/irrationalnumbe00manngoog#page/n100/mode/1up.

More Information

  • McAdams, David. Convergent. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Convergent.

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Absolute Convergence. 2010-01-04. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/absoluteconvergence.html.

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2010-01-04: Added "References" (McAdams, David.)
2008-11-19: Initial version (McAdams, David.)

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