PlanetMath: Leibniz' estimate for alternating series

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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 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

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

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

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 , 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 and $S_{m+1}$ for any . So the remainder after the 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, strictly, limit, Cauchy sequence, positive, negative, partial sums, series, sum, absolute value, remainder term, converges, alternating series
There are 4 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 3912 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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