9.1 Sequences & Series: Convergence & Divergence

Transcription

1 Notes 9.: Cov & Div of Seq & Ser 9. Sequeces & Series: Covergece & Divergece A sequece is simply list of thigs geerted by rule More formlly, sequece is fuctio whose domi is the set of positive itegers, or turl umbers, N =,,,. The rge of the fuctio re clled the terms i the sequece,, such tht { } Where,,,,, is clled the th term (or rule of sequece), d we deote the sequece by { } The sequece c be expressed by either ) mple umber of terms i the sequece, seprted by comms ) explicit fuctio defied by the rule of sequece ) the rule of sequece set off i brces. Exmple : The sequece,4,6,8, is the sequece of eve umbers. Express the sme sequece s rule of oegtive iteger. The sequece,,5, is the sequece of odd umbers. Express the sme sequece s rule of o-egtive iteger. How my i the list re eeded to estblish the rule i the bsece of the explicitly-stted rule?. ***NOTE: Whe give sequece s list, the first term is usully desigted to be ssocited with =. This is becuse we re usig s ordil (or coutig) umber, rther th crdil umber. We will be primrily iterested i wht hppes to the sequece for icresigly lrge vlues of. Exmple : 4 If = +, list out the first five terms, the estimte lim. Pge of 5

2 Notes 9.: Cov & Div of Seq & Ser FACT: be sequece of rel umbers. Let { } Possibilities: ) If lim ) If lim ) If lim 4) If lim =, the { } diverges to ifiity =, the { } diverges to egtive ifiity = c, fiite rel umber, the { } coverges to c oscilltes betwee two fixed umbers, the { } diverges by oscilltio Defiitio:! is red s fctoril. It is defied recursively s ( ) ( ) ( )( )( ) Por ejemplo: 9! = ! =! or s! =! = Exmple : Determie whether the followig sequeces coverge or diverge.,,, 4,,, (b),,,,,, (c) = + ( ) (d) = (e) l = (f) =!! ( + ) (g) =!! ( ) (h) ( ) + = (i) = ( ) ( ) (j) =! ( + ) (k) = + (l) ( )! Pge of 5

3 Notes 9.: Cov & Div of Seq & Ser Sometimes, lbeit rrely, we hve to write the rule of sequece s fuctio of from ptter. Exmple 4: Write expressio for the th term., 8,, 8,... (b) 5, -5, 45, -5,... (c), 4, 9, 6, 5,... (d) 4, 0, 8, 8,... (e),, 4 5, 5 7, 6,... (f) l, l, l 4, l8, 9 A Series is the sum of the terms i sequece. Fiite sequeces d series hve defied first d lst terms, wheres ifiite sequeces d series cotiue idefiitely. A series is iformlly the result of ddig y umber of terms from sequece together: A series c be writte more succictly by usig the summtio symbol sigm,., the Greek letter S for Esum (the E is both silet d ot relly there.) For ifiite series, we c look t the sequece of prtil sums, tht is, lookig to see wht the sums re doig s we dd dditiol terms. I geerl, the th prtil sum of series is deoted S. This c be explored o clcultor by ddig sequetil terms to the ggregte sum. Exmple 5: For both = d b =, geerte the sequece of prtil sums S, S, S,, S, for ech, the determie if the sequeces coverges or diverges. Do the results surprise you? Where else hve we see somethig like this before? Pge of 5

4 Notes 9.: Cov & Div of Seq & Ser Covergece d Divergece of Series Wht does it me for series to coverge? To diverge? Let s look t couple series from specil fmily clled geometric series. Exmple 6: Give the series = , = fid the first te terms of the sequece of prtil sums, d list them below, S, S, S,, S0. Bsed o this sequece of prtil sums, do you thik the series coverges? Diverges? To wht? (HINT: first rewrite the rule of sequece so tht it looks like expoetil fuctio of.) Exmple 7: Give the series = , fid the first five terms of the sequece of prtil = sums, d list them below. Bsed o this sequece of prtil sums, do you thik the series coverges? Diverges? To wht? Pge 4 of 5

5 Notes 9.: Cov & Div of Seq & Ser We re ow goig to look t severl fmilies of ifiite series d severl tests tht will help us determie whether they coverge or diverge. For some tht coverge, we might be ble to give the ctul sum, or itervl i which we kow the sum will be. For others, simply kowig tht they coverge will hve to suffice. Geometric Series, th Term Test for Divergece, d Telescopig Series Geometric Series Test (GST) A geometric series is i the form r or = 0 The geometric series diverges if r. = r, 0 If r <, the series coverges to the sum S =. r Where is the first term, regrdless of where strts, d r is the commo rtio. Exmple 8: Usig the GST, determie whether the followig series coverge or diverge. If the coverge, fid the sum. = (b) = (c) = th Term Test for Divergece (ONLY) If lim 0, the the series Note: This does NOT sy tht if lim = 0, the the series DOES coverge. This test c oly be used diverges. = (thik bout it, it should mke perfect sese!) to prove tht series diverges (hece the me.) If lim = 0, the this test does t tell us ythig, is icoclusive, does t work, fils, etc.... We MUST use other test. This test c be GREAT timesver. Alwys perform it FIRST, ot secod, but FIRST!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Pge 5 of 5

6 Notes 9.: Cov & Div of Seq & Ser Exmple 9: Determie whether the followig series coverge or diverge. If they coverge, fid their sum. + (b) 5 =! (c)! = + (d) = = ( ). A series such s is clled telescopig series becuse it collpses to oe 4 term or just few terms. If series collpses to fiite sum, the it coverges by the Telescopig Series Test. Exmple 0: Determie whether the followig series coverges or diverges. If they coverge, fid their sum. + + (b) = (c) ( ) = + = Pge 6 of 5

7 Notes 9.: Cov & Div of Seq & Ser Itegrl Test d p-series Itegrl Test If f is Decresig, Cotiuous, d Positive (Dogs Cuss i Priso!) for x AND = f ( x), the d ( ) = f x dx either BOTH coverge or diverge. Note : This does NOT me tht the series coverges to the vlue of the defiite itegrl!!!!!! Note : The fuctio eed oly be decresig for ll x > k for some k. If the series coverges to S, the the remider, R = S S is bouded by 0 R f ( x) dx. (Not o AP exm, but o my exm.). This mes S [ S, S R ] +. Exmple : Determie whether the followig series coverge or diverge. If they coverge, fid itervl i which the sum resides usig S 4. (b) = + = + Exmple : Approximte the sum of the coverget series mximum error for your pproximtio. by usig six terms. Iclude estimte of the 4 = Pge 7 of 5

8 Notes 9.: Cov & Div of Seq & Ser p-series A series of the form positive costt. = is clled p-series, where p is p p p p p = For p =, the series = is clled the hrmoic series. = Bsed o your experiece with p-series d their relice o the umber oe, fill i chrt below. p-series Test The p-series = , bsed o bove p p p p p = ) If p =, b) If p <, c) If p >, Note: If the p-series coverges, we cot fid its sum. This is more ofte the cse th ot. Exmple : Determie of the followig coverges or diverges: = (b) = (c) = (d) = Pge 8 of 5

9 Notes 9.: Cov & Div of Seq & Ser Bsed o your experiece with improper itegrls, gi, fill i the chrt below. Compriso of Series Direct Compriso Test (DCT) If 0 d b 0, ) If b coverges d 0 b, the =. = ) If diverges d 0 b =, the b. = NOTE:You must stte/show the iequlity whe sttig the coclusio of the test!! Exmple 4: Determie whether the followig coverge or diverge. (b) = + (c) = + = + (d) (e) = 4 = cos (f) 4 = 0 Pge 9 of 5

10 Notes 9.: Cov & Div of Seq & Ser Sometimes the iequlities eeded bove do t hold or re difficult to show, but you still strogly suspect the result becuse you recogize similr series with which to compre it. Limit Compriso Test (LCT) If 0 d b 0, d lim fiite d positive. The the two series b d = b = L or lim = L, where L is both b either both coverge or both diverge. = Exmple 5: Determie whether the followig coverge or diverge. (b) = = (c) (d) = = Pge 0 of 5

11 Notes 9.: Cov & Div of Seq & Ser Altertig Series A ltertig series is series whose terms re ltertely positive d egtive o cosecutive terms. 4 + For istce: + + ( ) d + + ( ) =!! 4!! I geerl, just kowig tht lim = 0 tells us very little bout the covergece of the series = ; = however, it turs out tht ltertig series must coverge if its terms cosistetly shrik i size d pproch zero!! Altertig Series Test (AST) >, the the ltertig series ( ) or ( ) If 0 Note: This does NOT sy tht if lim 0 the series DIVERGES by the AST. The AST c ONLY be used to prove covergece. If lim 0, the the series diverges, but by the th-term test NOT the AST. = of the followig coditios re stisfies: ) lim = 0 + coverges if both = is decresig (or No-icresig) sequece; tht is, + for ll > k, for some k Z ) { } Exmple 6: Determie whether the followig series coverge or diverge. = ( ) + = ( ) l ( ) (b) (c) = cos ( π ) (d) = ( ) (e)! ( ) ( ) (f) = 5 + = ( ) + (g) = Pge of 5

12 Notes 9.: Cov & Div of Seq & Ser ABOSOLUTE VS CONDITIONAL CONVERGENCE Theorem: If the series coverges, the = lso coverges. = Crzy Fct: Sometimes mere rerrgemet of terms i coverget ltertig series c yield differet sums!!! Such series is clled bsolutely coverget. Notice tht if it coverges o its ow, the ltertor oly llows it to coverge more rpidly. is coditiolly coverget if = coverges but = diverges. = Exmple 7: Determie whether the give ltertig series coverges or diverges. If it coverges, determie whether it is bsolutely coverget or coditiolly coverget. = ( ) (b) = ( ) + Alterte Series Remider Suppose ltertig series stisfies the coditios of the AST, mely tht lim = 0 d { } is ot icresig. If the series hs sum S, the R = S S +, where S is the th prtil sum of the series. I other words, if ltertig series stisfies the coditios of the AST, you c pproximte the sum of the series by usig the th prtil sum, S, d your error will hve bsolute vlue o greter th the first term left off, +. This mes S [ S R, S + R ] Exmple 8: ( ) Approximte the sum by usig its first six terms, d fid the error. Use your results to fid =! itervl i which S must lie. Pge of 5

13 Notes 9.: Cov & Div of Seq & Ser Exmple 9: Approximte the sum of = ( ) + with error less th Rtio d Root Tests Rtio Test Let.. be series of ozero terms. = coverges if = diverges if = + lim < + lim >. The rtio test is icoclusive if + lim = Series ivolvig expressios tht grow very rpidly such s fctorils d/or expoetil work especilly well with the Rtio Test. Exmple 0: Determie if the followig coverge or diverge. =! (b) = + (c) = ( + )! Pge of 5

14 Notes 9.: Cov & Div of Seq & Ser Root Test.. coverges if lim < = diverges if lim > =. The Root Test is icoclusive if lim = If the etire rule of sequece c be writte s power of, the Root Test is hrd to bet! Exmple : = e (b) = + 4 Exmple : Puttig it ll together. Determie if the followig series coverge or diverge. Nme the test used d the criteri of ech test used. (c) (e) (g) (i) (k) (m) (o) = + + (b) = 4 (d) (f) 5 = (h) = + 4 (j) = + (l) 5 + = 5 () = = + (p) = 0 7 = 5 = ( l ) 4 = = = cos π = = ( l ) ( ) l Pge 4 of 5

15 Notes 9.: Cov & Div of Seq & Ser To help you remember ll these tests, just thik of Moses fleeig with the Isrelites from Phroh: PARTING C P p-series: Is the series i the form p? A R T I N Altertig series: Does the series lterte? If it does, re the terms gettig smller, d is the th term 0? Rtio Test: Does the series coti thigs tht grow very lrge s icreses (expoetils or fctorils)? Telescopig series: Will ll but couple of the terms i the series ccel out? Itegrl Test: C you esily itegrte the expressio tht defies the series (re Dogs Cussig i Priso?) th Term divergece Test: Is the th term somethig other th zero? G Geometric series: Is the series of the form =0 r? C Compriso Tests: Is the series lmost other kid of series (e.g. p-series or geometric)? Which would be better to use: the Direct or Limit Compriso Test? Pge 5 of 5