1 Section 8.1: Sequences. 2 Section 8.2: Innite Series. 1.1 Limit Rules. 1.2 Common Sequence Limits. 2.1 Denition. 2.

Transcription

1 Clculus II d Alytic Geometry Sectio 8.: Sequeces. Limit Rules Give coverget sequeces f g; fb g with lim = A; lim b = B. Sum Rule: lim( + b ) = A + B, Dierece Rule: lim( b ) = A B, roduct Rule: lim b = AB, Costt Multiple: lim k = ka, Quotiet Rule: lim b = A B whe B 6= 0. Sdwich Thm: If c b for some sequece fc g d A = B, the A = lim c = B. Check: This does ot hold for diverget sequeces. Sequeces re esier whe cosiderig these rules. of sequece.2 Commo Sequece Limits l. lim = lim p =. 3. lim = = ( > 0). 4. lim = 0 (jj < ). 5. lim + = e. 6. lim = 0. of sequece 2 Sectio 8.2: Iite Series 2. Deitio Give iite sum, =0. =0 := lim m m! =0. There re lots of covergece tests, but this deitio will be importt for telescopig sums. series 2.2 th Term Test A sum, but 6! 0. If lim 6= 0, the diverges. Check: This oly works to show divergece, ot covergece! Lots of series c be elimited quickly this wy! Alwys try this rst! test 2.3 Geometric Series Give sum i the form =0 r. Necessry: Be creful of ide requiremets, d jrj <. =0 r = r. Check: Try few prtil sums to see if it coverges to the clculted result. This is commo emple. series

2 Clculus II d Alytic Geometry 2.4 Series Combitios Give two coverget series = A; b = B, the. Sum Rule: ( + b ) = + b = A + B. 2. Dierece Rule: ( b ) = = A B. 3. Costt Multiple Rule: k = k = ka. Kowig prts of sums rst my help occsiolly. Note tht this does NOT work whe the either or b diverge! 3 Sectio 8.3: Itegrl Test 3. Itegrl Test Cosider the sum =A d there eists fuctio f() with f() = for A. Necessry: f() 0, decresig. =A d R f()d behve the sme. Use covergece tricks from 7.7. A Some series c be checked usig improper itegrls esier th other tests. test 3.2 p-series A sum c be orgized s =. If p >, coverges. 2. If p, diverges. p. If p, the direct compriso test shows, so diverges. It turs out tht y lrger p power will coverge. series (mthemtics)#-series 4 Sectio 8.4: Compriso Tests 4. Direct Compriso Test (DCT) Showig covergece or divergece of sum whe c be compred to b d b is kow. Necessry: 0.. If b d b coverges, the coverges. 2. If b d b diverges, the diverges. test 4.2 Limit Compriso Test (LCT) Show covergece or divergece of sum =0 whe behves similrly to b d =0 b is kow to coverge or diverge.. If lim = c > 0, the d b behve the sme. 2. If lim = 0, d b coverges, the coverges. 3. If lim = +, d b diverges, the diverges. Sometimes this is esier compriso th the direct route. compriso test 2

3 Clculus II d Alytic Geometry 5 Sectio 8.5: Rtio d Root Tests 5. Rtio Test Cosider sum =0 where cosecutive terms + d hve my shred terms (th powers, fctorils, etc). Necessry: Let lim + =. The, () coverges if <, (b) diverges if >, (c) icoclusive whe =. This is prticulrly useful for power series. test 5.2 Root Test Cosider sum =0 where hs th-epoets roud complicted fuctios. Necessry:. 0 for N, 2. lim p =, The, () coverges if <, (b) diverges if >, (c) icoclusive whe =. Some series c oly be solved usig Root Test test 6 Sectio 8.6 Altertig Series, Absolute d Coditiol Covergece 6. Altertig Series Test Give series ( =0 ) Necessry: lim = 0. Whe ll three re true, the the sum coverges. Severl series coverge coditiolly, but ot bsolutely. See ltertig hrmoic series. series 6.2 Absolute/Coditiol Covergece Necessry: If series hs positive d egtive terms, it c coverge bsolutely (whe ll the terms re tke with bsolute vlue) or coditiolly (coverget oly whe the terms keep their sig) or lwys diverge (ever coverget series, o mtter wht the sigs do). Check coditiol covergece with the Altertig Series Test, covergece 7 Sectio 8.7: ower Series 7. ower Series Wht: A power series bout hs the form =0 c ( ) for sequece c. ower series bsolutely coverge for vlues of give by the Rtio Test. Check boudry coditios for coditiol covergece. ower series re importt to uderstd before tryig Tylor series. series of covergece 3

4 Clculus II d Alytic Geometry 7.2 Term-by-Term Itegrtio d Derivtio Approimte fuctio by kowig power series for derivtive or itergrl. d d =0 c ( ) = R = c ( ) : [ =0 c ( ) ] d = c =0 ( + )+ + C. Try provig d d e = e usig its Tylor series. series#dieretitio d itegrtio 8 Sectio 8.8: Tylor d Mcluri Series 8. Tylor/Mcluri Series Approimtig or simplifyig epressios for fuctio f() by usig polyomil epressio. Necessry: f() iitely dieretible o itervl ( R; + R), d kowig the vlues of ech derivtive t. Check: () = k=0 f (k) () ( ) k is the th Tylor olyomil. The limit of this polyomil k! f () () ( ). s pproches iity is the Tylor Series: =0 The Mcluri series is the Tylor series with = 0. Crefully wtch the derivtives d the ptter creted. olyomils re esier to do lot of thigs, icludig derivte d itegrte. series 9 Sectio 8.9: Covergece of Tylor Series 9. Tylor's Theorem d Formul Necessry: A fuctio f() cotiuous d ( + )-dieretible o itervl ( R; + R). The Tylor polyomil () d the remider R () sum to ectly the fuctio: f() = () + R () where R () = f + (c) ( (+)! )+ for some c strictly betwee d. A elborte use of the Itermedite Vlue Theorem, pired with the remider estimtio theorem proves covergece of Tylor series. 9.2 Remider Estimtio Theorem Necessry: Check: Computig how close Tylor polyomil of degree will estimte the fuctio. Kowig () d the ( + )th derivtive. Give Tylor polyomil () cetered t, to be clculted t, let M be upper boud for jf (+) (t)j for < t <. The, the remider term R () stises jr ()j j j+ (+)! M Clculte the polyomil d the fuctio vlues t, d see if they re s close s this estimte. Computers c clculte polyomils better th geerl fuctios, but it would be good to kow how my computtios re required to chieve certi degree of ccurcy. polyomil#estimtes of the remider 9.3 Frequetly Used Tylor Series 4

5 Clculus II d Alytic Geometry Askig to pproimte swer usig \kow" Tylor series. Necessry: Clcultig vlue usully hrd to compute by hd. Geometric: = = =0 ; jj < Altertig Geometric: + = ( ) + = ( =0 ) ; jj < Epoetil: e = = 2! 3! =0 ; jj < Sie: si = ( ) + = 2+ 3! 5! (2+)! =0 ( ) ; jj < (2+)! Cosie: cos = ( ) + = 2 2! 4! (2)! =0 ( ) ; jj < (2)! Nturl Log: l( + ) = ( 2 3 ) + = ( = ) ; < ( ) Nturl Log: l() = ( ) 2 ( ) ( ) ( ) = ( = ) ( ) ; 0 < 2 Nturl Log: l + = 2 th = = =0 ; jj < 2+ Iverse Tget: t = ( ) + = =0 ( ) ; jj < 2+ Check: Compute the rst few derivtives d check their vlue. Mke sure this grees with your rst few vlues. Note tht zero vlues re removed from these sums. It is esier to memorize few emples th to repetedly geerte them. series#list of Tylor series of some commo fuctios 0 Sectio 8.0: Biomil Series 0. Deitio Lookig for Tylor series represettio of f() = ( + ) m Note tht f (k) () = m k k m k = m(m )(m 2) (m k+) The f() = k! m k=0 k k. Kowig this epsio mkes polyomil epsio esier. series 5