MATH 1272 Lecture Notes - Lecture 1: Conditional Convergence, Ratio Test
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We say series an is absolutely convergent (or converges absolutely) if series |an| converges. We say series is conditionally convergent if series an converges but series|an| diverges. If series an is absolutely convergent, then series an converges. For all n, and 2an converges to 2 |an| (since |an| converges) So (an+ |an|) converges by comparison test, and since |an| converges also, ((an +|an|)-|an|) = an converges. So by comparison test, the original series converges. If l > 1 (including l = ), then an diverges. If l=1 or dne but not , the to is inconclusive (we don"t know) If l >1 (including l = ), then an diverges. If l =1 or dne but not , the test is inconclusive. So, our original series converges absolutely by the ratio test. By dominating terms, which we can use since it"s just a limit as n (cid:736) . Do, 2/3 < 1, so our original converges absolutely by the root test.
