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[parent] Leibniz' estimate for alternating series (Theorem)
Theorem 1   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 as the first omitted term $ \pm p_{m+1}$ and the absolute value less than $ p_{m+1}$.

Proof. The convergence of (1) is proved here. 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 term $ -p_{2n}$. Then the remainder term

$\displaystyle R_{2n} = S\!-\!S_{2n}$
may be written in the form

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

$\displaystyle R_{2n} = p_{2n+1}-(p_{2n+2}-p_{2n+3})-(p_{2n+4}-p_{2n+5})-\ldots$
The former shows that $ R_{2n}$ is positive as the first omitted term $ p_{2n+1}$ and the latter that $ \vert R_{2n}\vert < p_{2n+1}$. Similarly one can see the assertions true when the series (1) is truncated after a positive term $ p_{2n-1}$.


A pictorial proof.

\includegraphics{leibniz-sequence.eps}

As seen in this diagram, whenever $ m' > m$, we have $ \lvert S_{m'}\!-\!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 centre of the spiral, strictly in between $ S_m$ and $ S_{m+1}$ for any $ m$. So the remainder after the $ m$th term must have the same direction as $ \pm p_{m+1} = S_{m+1}\!-\!S_m$ and lesser magnitude.

Example 1. The alternating series

$\displaystyle \frac{1}{\sqrt{2}-1}-\frac{1}{\sqrt{2}+1} +\frac{1}{\sqrt{3}-1}-\frac{1}{\sqrt{3}+1} +\frac{1}{\sqrt{4}-1}-\frac{1}{\sqrt{4}+1}+-\ldots$
does not fulfil the requirements of the theorem and is divergent.

Example 2. The alternating series

$\displaystyle \frac{1}{\ln{2}}-\frac{1}{\ln{3}}+\frac{1}{\ln{4}}-\frac{1}{\ln{5}}+-\ldots$
satisfies all conditions of the theorem and is convergent.



"Leibniz' estimate for alternating series" is owned by pahio. [ full author list (2) ]
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See Also: e is irrational

Other names:  Leibniz' estimate for remainder term

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Cross-references: convergent, divergent, theorem, strictly, limit, Cauchy sequence, diagram, positive, negative, partial sums, series, sum, proof, absolute value, remainder term, converges, alternating series
There are 5 references to this entry.

This is version 30 of Leibniz' estimate for alternating series, born on 2004-11-24, modified 2007-12-02.
Object id is 6523, canonical name is LeibnizEstimateForAlternatingSeries.
Accessed 5556 times total.

Classification:
AMS MSC: 40-00 (Sequences, series, summability :: General reference works )
 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
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