Absolute Convergence

Pronunciation: /æb səˈlut kənˈvɜrdʒ əns/ Explain

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. Merriam-Webster. Last Accessed 8/6/2018. https://www.merriam-webster.com/dictionary/absolute%20convergence.
  2. Manning, Henry Parker, Ph.D.. Irrational Numbers and Their Representation by Series and Sequences. pg 89. www.archive.org. John Wiley & Sons. 1906. Last Accessed 8/6/2018. http://www.archive.org/stream/irrationalnumbe00manngoog#page/n100/mode/1up. Buy the book

More Information

  • McAdams, David E.. Convergent. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/19/2018. http://www.allmathwords.org/en/c/convergent.html.
  • Conditional & absolute convergence. www.khanacademy.org. Kahn Academy. 6/19/2018. https://www.khanacademy.org/math/ap-calculus-bc/bc-series/bc-ratio-alt-series/v/conditional-and-absolute-convergence.

Cite this article as:

McAdams, David E. Absolute Convergence. 6/12/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/absoluteconvergence.html.

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Revision History

6/12/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
1/4/2010: Added "References". (McAdams, David E.)
11/19/2008: Initial version. (McAdams, David E.)

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