powered by Google (TM)
index: click on a letter
A B C D E
F G H  I  J
K L M N O
P Q R S T
U V W X Y
Z A to Z index
index: subject areas
numbers & symbols
sets, logic, proofs
geometry
algebra
trigonometry
advanced algebra
& pre-calculus
calculus
advanced topics
probability &
statistics
real world
applications
multimedia
entries
about mathwords  
website feedback  


Alternating Series Remainder

A quantity that measures how accurately the nth partial sum of an alternating series estimates the sum of the series. If an alternating series is not convergent then the remainder is not a finite number.

Consider the following alternating series (where ak > 0 for all k) and/or its equivalents.

\[\sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^{k + 1}}{a_k}} = {a_1} - {a_2} + {a_3} - {a_4} + \cdots \]

If the series converges to S, then the nth partial sum Sn and the corresponding remainder Rn can be defined as follows.

\[{S_n} + {R_n} = S\] \[{S_n} = \sum\limits_{k = 1}^n {{{\left( { - 1} \right)}^{k + 1}}{a_k}} \] \[{R_n} = \sum\limits_{k = n + 1}^\infty {{{\left( { - 1} \right)}^{k + 1}}{a_k}} \]

This gives us the following

\[{R_n} = S - \sum\limits_{k = 1}^n {{{\left( { - 1} \right)}^{k + 1}}{a_k}} \]

If the series converges to S by the alternating series test, then the remainder Rn can be estimated as follows for all nN:

\[\left| {{R_n}} \right| \le {a_{n + 1}}\]

Note that the alternating series test requires that the numbers a1, a2, a3, ... must eventually be nonincreasing. The number N is the point at which the values of an become non-increasing.

anan +1 for all nN, where N ≥ 1.

 

See also

Remainder of a series, convergence tests, divergent series

 


  this page updated 19-jul-17
Mathwords: Terms and Formulas from Algebra I to Calculus
written, illustrated, and webmastered by Bruce Simmons
Copyright © 2000 by Bruce Simmons
All rights reserved
NCTM Web Bytes December 2004 Web Bytes March 2005 Web Bytes