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Leibniz' estimate for alternating series
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(Theorem)
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Proof. The convergence of (1) is proved here. Now denote the sum of the series by and the partial sums of it by
. Suppose that (1) is truncated after a negative term . Then the remainder term
may be written in the form
or
The former shows that is positive as the first omitted term and the latter that
. Similarly one can see the assertions true when the series (1) is truncated after a positive term .
A pictorial proof.
As seen in this diagram, whenever , we have
. Thus the partial sums form a Cauchy sequence, and hence converge. The limit lies in the centre of the spiral, strictly in between and for any . So the remainder after the th term must have the same direction as
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 convergent.
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"Leibniz' estimate for alternating series" is owned by pahio. [ full author list (2) ]
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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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Pending Errata and Addenda
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