is absolutelyconvergent,thenit is convergent.
The proofis quitestraightforwardand is left as an exercise.The converseof Theorem5.1.1is not true. The
series is convergentby the Al- ternatingSeriesTest, but its absoluteseries, (the
harmonicseries),is divergent.
Example 1 is absolutelyconvergentsince for any 1 ≤n,θ, and is
convergent(e.g. by thep- test).Indeed,by the Integraland Comparisontests, is absolutely
convergentfor anyθandp> 1.
The limit of the ratio givesus a comparisonof the tail part (i.e ∑n Largean) of the series∑an
with a geometricseries.
Theorem(The RatioTest) Let ∑anbe a seriesof non-zeronumbers*.
(A) If ,thenthe seriesis absolutelyconvergent.
(B) If or then the seriesis absolutelydivergent.
(C) If then the test is inconclusive.
*: we couldignorethe zero-valuedan's as far as the sum is concerned.
Proof.(A) The proofis by comparisonwith a geometricseries.
If α < 1, then It followsfrom the definitionof limit that thereis an integerN,
for alln≥N. Let Then|aN+ 1| < β|aN|,|aN+ 2| < β|aN+ 1| < β^2 |aN|,... and recursivelywe
have|an+ 1| < βn-N+ 1|aN| forn≥N, and whichis finite.Combining
with the finitelymanyterms, is still finite.
(B) A similarargumentconcludes So the seriesis divergent.
Example 2 Test the series for absoluteconvergencewhereAis a constant.