Series Tests Reference Sheet

Series Tests Reference Sheet - Series Tests Reference Sheet...

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Unformatted text preview: Series Tests Reference Sheet Test fer Divergenee If Z en is e. series then we have twe eenditiens en {en}: 1. If lirn :1,1 = II], then the series see either reeverge er diverge. In this case we ge en te fl—Iflfl ether tests. ‘2. Il' liIn en 95 II}1 then Zen diverges. We ere finished with the prehlem. Ill—“W lGeernetrie Series If the series has the term 2 ernL where e riees net ehenge and r is the retier then we have the fellewing: 1. ll |r| E i, then the series diverges. ' . first term 2. If M s: i. then the series eenverges, end the sure 11-: Ear-“‘1 = — l—r Integral Test m Given a series 2 an 0f Pflfiiti‘m decreasing terms. if yeu can find a funetien fie} such that e=1 e11 = fin] fer e11 rt, then yeu ereste f fields. We knew thet the series end the integral 1 either heth diverge er beth eenverge- Nets: yen must check that HI] is in feet deereesieg...yeu usually de this vie. the first derivetive test. A eensequenee ef the integral test is the . m 1 p—Series Test: Given e series ei the ferrri E E then “ml 1. The series eenverges if n :e l. 2. The series diverges if i] 4:: p {i 1 {end teehnieelly fer all p E l]. Netiee that this test seems simiiiir te the geemetrie series. lmt is much different. Direet Cernperisen Test Let Z an be a series with he negative terms. Then: 1. X en eenverges if there is s eenvergeet EE'l'iE Z .21 with efl 5 en fer eli rt. '2. Zen diverges if there is a. divergent series 2 [in with e.” E d“ fer ell n. Limit Cemperisen Test Seppese that en s D and ti... 3: fl Fer ell n. Then: If [he — = c :a- [l and is finiter then 2 s" and Z 11,. either heth cenverge er heth diverge. Usually, veu are given Xe“. "fee must pick at suitshle Zen that yen siresti},r knew cenverges er diverges hv serne ether test {like the p—series test er test fer divergence etc.) Ratie Test Let 2 he, he s series with enthr pesitive terms. else. suppese thst Then we have the feliewing three cenditiens en L: 1. If L s: 11 then the series cenvErges. 2. If L is 1. then the series diverges. 3. if L = i, then the test is incenclusive. we must try seine ether reesening. Ahseinte |Cenvergerice Test If E |efl| cenverges then we have thst E eT1 else cenverges. {Nete: this says nething when 2 Ian] diverges. We must use snether test in this sitnstienr usually the Alternating Series Test er the Test [er Divergence-) Alternating Series Test IGiven a. series ef the term XII—llnflnn. This series will cenverge if the feliewing cenditiens held [All must held]: 1. The en's are all pesitive. 2. After serne eeint1 the sequence {11"} is decreasing, (is. n" 3 en”). 3. lift]: 111.I =- i} as e Sequence- 31""“3? Rent Test We use the rent test if the series invelves nth [sewers and is net 3 geemetric series. If Zen is e series then we cempnte e = lire f/lnfll- Then the series Z en will cenverge ehselutelv rs—u-ee- if D s": 1, will diverge if H I: 11 and the test is incenclusive if D = l. ...
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This note was uploaded on 04/17/2008 for the course MATH 10360 taught by Professor Edgar during the Spring '08 term at Notre Dame.

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Series Tests Reference Sheet - Series Tests Reference Sheet...

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