Leibniz’ estimate for alternating series

Theorem (Leibniz 1682).   If  $p_{1}>p_{2}>p_{3}>\cdots$  and  $\displaystyle\lim_{m\to\infty}p_{m}=0$,  then the alternating series

 $\displaystyle p_{1}-p_{2}+p_{3}-p_{4}+-\ldots$ (1)

converges.  Its remainder term has the same sign (http://planetmath.org/SignumFunction) as the first omitted $\pm p_{m+1}$ and the absolute value less than $p_{m+1}$.

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 $S_{1},\,S_{2},\,S_{3},\,\ldots$.  Suppose that (1) is truncated after a negative $-p_{2n}$.  Then the remainder term

 $R_{2n}\;=\;S\!-\!S_{2n}$

may be written in the form

 $R_{2n}\;=\;(p_{2n+1}-p_{2n+2})+(p_{2n+3}-p_{2n+4})+\ldots$

or

 $R_{2n}\;=\;p_{2n+1}-(p_{2n+2}-p_{2n+3})-(p_{2n+4}-p_{2n+5})-\ldots$

The former shows that $R_{2n}$ is positive as the first omitted $p_{2n+1}$ and the latter that  $|R_{2n}|.  Similarly one can see the assertions true when the series (1) is truncated after a positive $p_{2n-1}$.

A pictorial proof.

As seen in this diagram, whenever  $m^{\prime}>m$,  we have  $\lvert S_{m^{\prime}}\!-\!S_{m}\rvert\leq p_{m+1}\to 0$.  Thus the partial sums form a Cauchy sequence, and hence converge.  The limit lies in the of the spiral, strictly in $S_{m}$ and $S_{m+1}$ for any $m$.  So the remainder after the $m$th must have the same direction as  $\pm p_{m+1}=S_{m+1}\!-\!S_{m}$  and lesser magnitude.

Example 1.  The alternating series

 $\frac{1}{\sqrt{2}-1}-\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}-1}-\frac{1}{\sqrt{% 3}+1}+\frac{1}{\sqrt{4}-1}-\frac{1}{\sqrt{4}+1}+-\ldots$

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

Example 2.  The alternating series

 $\frac{1}{\ln{2}}-\frac{1}{\ln{3}}+\frac{1}{\ln{4}}-\frac{1}{\ln{5}}+-\ldots$

satisfies all conditions of the theorem and is convergent.

Title Leibniz’ estimate for alternating series LeibnizEstimateForAlternatingSeries 2014-07-22 15:34:38 2014-07-22 15:34:38 pahio (2872) pahio (2872) 35 pahio (2872) Theorem msc 40A05 msc 40-00 Leibniz’ estimate for remainder term EIsIrrational2 ConvergingAlternatingSeriesNotSatisfyingAllLeibnizConditions