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


may be written in the form




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


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

Example 2.  The alternating series


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