MATH 166 TEST 3 REVIEW SHEET

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1 MATH 66 TEST REVIEW SHEET Geeral Commets ad Advice: The studet should regard this review sheet oly as a sample of potetial test problems, ad ot a ed-all-be-all guide to its cotet. Aythig ad everythig which we have discussed i class is fair game for the test. The test will cover roughly Sectios 0., 0., 0., 0.4, ad 0.. Do t limit your studyig to this sheet; if you feel you do t fully uderstad a particular topic, the try to do more problems out of your textboo! Summary of Relative Growth Rates. For two sequeces a ) ad b ) such that a ) ad b ), let the symbols a ) << b ) deote that b ) grows faster tha a ). The the followig is a summary of our most commoly ecoutered growth rates: l ) << l ) r ) << q ) << ) << p ) << b ) <<!) << ), where r >, 0 < q <, p >, ad b >. Geeral Strategy for Determiig Covergece of a Series. There is o hard ad fast rule for determiig whether a series coverges or diverges, ad i geeral the studet should build up their ow ituitio o the subject through practice. However, the followig metal checlist ca be a helpful guidelie. ) Divergece Test. Give a series a, the first thig oe should always as is, do the terms a ) 0? If ot, we are doe ad the series diverges by the DT. If the terms do shri to zero, however, we have to put more thought ito the problem. ) Very Nice Series. Is the series a p-series with p >, or a geometric series, or a telescopig series? If so we are doe ad the series coverges by our geeral theorems. Note that sometimes a telescopig series ca be hidde : for example is a telescopig series i disguise. The + ) method of partial fractios may reveal these.) ) Alteratig Series Test. If the series is alteratig ad the terms go to zero, the the series coverges by the AST. 4) Itegral Test. Do the terms of the series loo lie a fuctio we could probably itegrate? As a example, the series + shouts Substitutio Rule whe we cosider its improper itegral aalogue ) x + x ) dx. Remember the series ad the improper itegral are ot equal to oe aother, but they do either both coverge or both diverge.) ) Direct Compariso Test ad Limit Compariso Test. Do the terms a of the series behave similarly to those of oe of the Very Nice Series? If

2 8 so, a compariso test may be appropriate. Usually the LCT is more powerful tha the DCT. For examples, the series + seems to 7 behave lie for very large ad so we thi it probably coverges; coversely the series seems to behave lie a costat multiple of the harmoic series for very large, so we thi it diverges. Usually the atural choices for a compariso test are a p-series or a geometric series.) 6) Ratio Test. Are we looig at a series of fractios where the questio of covergece seems to come dow to comparig the growth rate of the umerator vs. the deomiator? e.g. or l ) 4 or 000.! The Ratio Test has a good chace of worig. Ratio Test is almost always the most atural choice if there is a factorial! term appearig aywhere i the series.) 7) Root Test. Are we looig at a series where to the -th power occurs ) i a sigificat way? e.g. or + ). Root Test + =0 might be easiest. 8) Words of Warig. Your first attempt at a test may ot be the right choice. Be aware of whe a test fails for istace if the limit i Ratio/Root Test is exactly, or extremal cases of Limit Compariso Test where limit is 0 or, etc.) ad be prepared to try a differet strategy. O that ote, it is ofte very helpful to do some private scratchwor where you determie for yourself if a particular test solves the problem, ad the carefully write a solutio for full credit i a homewor, quiz, or exam settig.. Determie if the sequece coverges, ad if so, state to what limit. a) 4 b) arcta000) c) e 000 = d) + e) f) arcta 7 7

3 g) a =, where a = 0 ad a = a + 0 for every h) b = where b is the -th digit after the decimal poit i the decimal expasio of π i) + ) ), where is a fixed real costat.. Compute the followig series, if they coverge. a) b) c) =0 = e ) = + 6) + 7) d) e) f) g) ) 000 = 6 + =0. State whether the followig series coverge or diverge. Carefully justify each aswer. a) l ) 0 =0! b) =0 c) e d) e) f) g) 4 l ) ) = =

4 4 h) i) j) )!) )! = 00 + =0 ) = 8 + ) 4. Determie whether the followig series diverge, coverge absolutely, or coverge coditioally. ) a) b) c) d) e) f) g) = = = ) si ) + 0 = = )! ) / + /

5 Aswer Key Disclaimer: This ey was writte quicly ad may cotai errors or typos! Please let me ow if you have detected oe.. a. b. π c. diverges d. diverges e. 0 f. 0 g. h. diverges sice π is irratioal! i. e. a. e e b. 0 c. 7 d. 6 e. diverges f. 4 g. 6. a. diverges Sugg: Divergece Test) b. coverges Sugg: Ratio Test) c. coverges Sugg: Itegral Test) d. coverges Sugg: Direct Compariso Test) e. coverges Sugg: Root Test) f. diverges Sugg: Limit Compariso Test) g. diverges Sugg: Limit Compariso Test) h. coverges Sugg: Ratio Test) i. coverges Sugg: Root Test) j. coverges Sugg: Root Test). coverges Sugg: Direct Compariso Test) 4. a. coverges coditioally Sugg: AST, ad p-series) b. coverges absolutely p-series) c. coverges absolutely Sugg: DCT) d. diverges Terms do t coverge to 0!) e. coverges absolutely geometric series) f. coverges absolutely Sugg: Ratio Test) g. coverges coditioally Sugg: AST, ad LCT with p-series)