PlanetMath: converging alternating series not satisfying all Leibniz' conditions

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converging alternating series not satisfying all Leibniz' conditions

(Example)

The alternating series

$\displaystyle \sum_{n=1}^\infty\frac{(-1)^{n-1}}{n\!+\!(-1)^{n-1}} \;=\; \frac{1}{2}-\frac{1}{1}+\frac{1}{4}-\frac{1}{3}+\frac{1}{6}-\frac{1}{5}+-\ldots$

(1)

satisfies the other requirements of Leibniz test except the monotonicity of the absolute values of the terms. The convergence may however be shown by manipulating the terms as follows.

We first multiply the numerator and the denominator of the general term by the difference $n\!-\!(-1)^{n-1}$ , getting from (1)

$\displaystyle \sum_{n=1}^\infty\frac{(-1)^{n-1}}{n\!+\!(-1)^{n-1}} \;=\; \frac{... ...sum_{n=2}^\infty\left(\frac{(-1)^{n-1}n}{n^2\!-\!1}-\frac{1}{n^2\!-\!1}\right).$

(2)

One can state that the series

$\displaystyle \sum_{n=2}^\infty\frac{(-1)^{n-1}n}{n^2\!-\!1}$

(3)

satisfies all requirements of Leibniz test and thus is convergent. Since $$ 0 \;<\; \frac{1}{n^2\!-\!1} \;<\; \frac{1}{n^2\!-\!\frac{1}{2}n^2} \;=\; 2\cdot\frac{1}{n^2}\quad\mbox{for}\quad n \geqq 2, $$ and the over-harmonic series $\sum_{n=2}^\infty\frac{1}{n^2}$ converges, the comparison test guarantees the convergence of the series

$\displaystyle \sum_{n=2}^\infty\frac{1}{n^2\!-\!1}.$

(4)

Therefore the difference series of (3) and (4) and consequently, by (2), the given series (1) is convergent.

"converging alternating series not satisfying all Leibniz' conditions" is owned by pahio.

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See Also: sum of series depends on order

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Cross-references: comparison test, converges, over-harmonic series, convergent, series, difference, denominator, numerator, terms, absolute values, monotonicity, Leibniz test, alternating series

This is version 3 of converging alternating series not satisfying all Leibniz' conditions, born on 2009-08-27, modified 2009-08-28.
Object id is 11882, canonical name is ConvergingAlternatingSeriesNotSatisfyingAllLeibnizConditions.
Accessed 250 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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