MA241-012, Fall 2008Test 4 review sheetDisclaimer: “This review sheet is only intended to give you some guidance, and thus should not be taken asan exhaustive listing of possible test question topics. Do not expect this every time.”Calculators will not be allowed on this test. No notes or other aids will be allowed on this test.The test will cover §§8.2-8.5§8.2You need to know what a series is, what it means for a series to converge/diverge (and how that is differentfrom a sequence converging/diverging). You need to know the n-th term test for divergence. You need to beable to recognize a geometric series and determine if it is divergent or convergent (and if convergent, whatit converges to). You need to be able to recognize a telescopic series, determine if it is convergent, and, ifso, what it converges to. You need to be able to express a repeating decimal as a ratio of two integers (i.e.,as a fraction).§§8.2-8.4You need to know and be able to use all of the series convergence tests discussed in these sections (we talkedabout all of them in class): nth-term test for divergence, harmonic series, telescopic series, p-series, integraltest, direct comparison test (DCT), limit comparison test (LCT), alternating series test (AST), absoluteconvergence test (ACT), and the ratio test. On the exam, you will not be told which test to use on aparticular problem .You need to be sure to state which test you’re using, and no matter which test you use, you needto show that all conditions on the terms are satisfied before you try to use a test. For instance, theintegral test, DCT, and LCT all require that the series have positive terms, the integral and alternatingseries tests require that the terms (non-alternating part in AST) be decreasing, and the inequality in theDCT needs to be going the right way.Also remember that sometimes you need tests within tests, and so you may have to show that condi-tions are satisfied for multiple tests in the same problem. For example, in both comparison tests, you needto know about the convergence or divergence of the series you’re using for the comparison; in the absoluteconvergence test, you need to be able to show the convergence of the absolute value of the terms.As you go through the homework, try to be aware of the types of series that you’re using the differenttests on (for example, series involving n! and anoften work out nicely with the ratio test). If you’re stillhaving some difficulty decided which to use, James Cook (one of the other grad students) put together aflow chart which you may find useful: http://www4.ncsu.edu/∼jscook3/ma241conv div tests 204 210.pdf(see the last page). Note that he did not cover the DCT or LCT in that class - but you willdefinitely need to use these on the test.1Finally, be sure you know when a test is inconclusive. For instance:• The nth-term test is inconclusive if limn→∞an= 0.• The DCT is inconclusive if you have (your series)≤(divergent series) or (convergent series)≤(yourseries).• The LCT is inconclusive if limn→∞anbn= 0 or ∞.• The AST is inconclusive if bnis not decreasing or positive.• The ACT is inconclusive ifP∞n=1|an| (the absolute value of the terms) is divergent (for example, weknowP∞n=1(−1)nnis convergent by the AST. But, if you try to use the ACT and take the absolutevalue of the terms, you getP∞n=11n, which is divergent, so the ACT is inconclusive).• The Ratio Test is inconclusive if limn→∞|an+1an| = 1If you get an inconclusive result in a problem on the exam, then you need to try another test - it will b epossible to conclude either “converges” or “diverges” for every problem on the exam.§8.5Given a power seriesP∞n=0cn(x − a)n, you need to be able to both the interval of convergence (IOC) andthe radius of convergence (R): we do this by setting up the ratio test, see which values of x make the limit< 1, and then check the endpoints if needed (rememb er you need to use the other convergence tests to checkthe endpoints - the ratio test can’t tell what happens at the endpoints). Remember that the IOC will alwayshave at least one point: a (since every powers series converges at the point at which it is centered). Theorem3 on page 596 and the following paragraph go through the three possible cases that may come up whenfinding the IOC (also see your notes from 11/17 for some additional comments I made when I wrote thistheorem on the board). You don’t need to know the statement of this theorem, but you should be aware ofall the possibilities that can