Leibniz’ estimate for alternating series


Theorem (Leibniz 1682).   If  p1>p2>p3>  and  limmpm=0,  then the alternating seriesMathworldPlanetmath

p1-p2+p3-p4+- (1)

converges.  Its remainder term has the same sign (http://planetmath.org/SignumFunction) as the first omitted ±pm+1 and the absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath less than pm+1.

Proof.  The convergence of (1) is proved here (http://planetmath.org/ProofOfAlternatingSeriesTest).  Now denote the sum of the series by S and the partial sums of it by S1,S2,S3,.  Suppose that (1) is truncated after a negative -p2n.  Then the remainder term

R2n=S-S2n

may be written in the form

R2n=(p2n+1-p2n+2)+(p2n+3-p2n+4)+

or

R2n=p2n+1-(p2n+2-p2n+3)-(p2n+4-p2n+5)-

The former shows that R2n is positive as the first omitted p2n+1 and the latter that  |R2n|<p2n+1.  Similarly one can see the assertions true when the series (1) is truncated after a positive p2n-1.


A pictorial proof.

As seen in this diagram, whenever  m>m,  we have  |Sm-Sm|pm+10.  Thus the partial sums form a Cauchy sequence, and hence converge.  The limit lies in the of the spiral, strictly in Sm and Sm+1 for any m.  So the remainder after the mth must have the same direction as  ±pm+1=Sm+1-Sm  and lesser magnitude.

Example 1.  The alternating series

12-1-12+1+13-1-13+1+14-1-14+1+-

does not fulfil the requirements of the theorem and is divergent.

Example 2.  The alternating series

1ln2-1ln3+1ln4-1ln5+-

satisfies all conditions of the theorem and is convergentMathworldPlanetmath.

Title Leibniz’ estimate for alternating series
Canonical name LeibnizEstimateForAlternatingSeries
Date of creation 2014-07-22 15:34:38
Last modified on 2014-07-22 15:34:38
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 35
Author pahio (2872)
Entry type Theorem
Classification msc 40A05
Classification msc 40-00
Synonym Leibniz’ estimate for remainder term
Related topic EIsIrrational2
Related topic ConvergingAlternatingSeriesNotSatisfyingAllLeibnizConditions