converging alternating series not satisfying all Leibniz’ conditions

The alternating seriesMathworldPlanetmath

n=1(-1)n-1n+(-1)n-1=12-11+14-13+16-15+- (1)

satisfies the other requirements of Leibniz test except the monotonicity of the absolute valuesMathworldPlanetmathPlanetmathPlanetmath 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)

n=1(-1)n-1n+(-1)n-1=12+n=2n-(-1)n-1n2-1(-1)n-1=12+n=2((-1)n-1nn2-1-1n2-1). (2)

One can that the series

n=2(-1)n-1nn2-1 (3)

satisfies all requirements of Leibniz test and thus is convergentMathworldPlanetmathPlanetmath.  Since

0<1n2-1<1n2-12n2= 21n2forn2,

and the over-harmonic series n=21n2 converges, the comparison testMathworldPlanetmath guarantees the convergence of the series

n=21n2-1. (4)

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

Title converging alternating series not satisfying all Leibniz’ conditions
Canonical name ConvergingAlternatingSeriesNotSatisfyingAllLeibnizConditions
Date of creation 2013-03-22 19:00:45
Last modified on 2013-03-22 19:00:45
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Example
Classification msc 40-00
Classification msc 40A05
Related topic SumOfSeriesDependsOnOrder
Related topic LeibnizEstimateForAlternatingSeries