Solution.Let. Then asn→ ∞. So by the RatioTest,
the seriesis absolutelyconvergentfor any constantA. Indeed,the sum is eAwhichis very largefor large
A, but still finite.
We see the limitationof the RatioTest is when doesnot exist (not ∞) or is 1.
Example 3 (RatioTest inconclusive) , and
for both and The former(harmonicseries)divergeswhilethe latterconverges(by, say, the
p - test).
Questions(relatedto the RatioTest) Whatif
- limit of existseparatelyfornodd andneven,i.e existbut are
different? - exist but are different?
Exercise
Determinewhetherthe followingseriesis absolutelyconvergent,conditionallyconvergent,or divergentwith
the RatioTest and othertestsif necessary:
1.
2.
3.
4.
nth- root Test
If the generalterm anresemblesan exponentialexpression,the followingtest is handy.
Theorem(The RootTest)
(A) If , then the seriesis absolutelyconvergent.