Infinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex:

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1 Ifiite Series Sequeces: A sequece i defied s fuctio whose domi is the set of positive itegers. Usully it s esier to deote sequece i subscript form rther th fuctio ottio.,, 3, re the terms of the sequece d is the th term Listig Terms of Sequece. { } {3 + (-) } b. { b } { } c. recursively defied sequece { c }, where c 5 d c + c 5. Ptter Recogitio for Sequeces E: Fid the sequece { } first five terms re RST, 3, 5, 7, 9 UVW,... E: Determie the th term of the sequece whose first five terms re RST, 8, 6 8 6, 4, 4,.. UVW Limit of Sequece If the terms of sequece pproch limitig vlue the sequece is sid to coverge. Def of the Limit of Sequece Let L be rel umber. The limit of sequece { } is L, writte s lim L if for ech ε >, there eists M > such tht L <ε wheever > M. If the limit L of sequece eists, the the sequece coverges to L. If the limit of sequece does ot eist, the the sequece diverges. Limit of Sequece Let L be rel umber. Let f be fuctio of rel vrible such tht lim f( ) If { } is sequece such tht f() for every positive iteger, the lim L L

2 E: Fid the limit of the sequece whose th term is Properties of Limits of Sequeces Let lim L d lim b. lim( ) ± b L± K. lim c 3. lim b K LK 4. lim b F I + HG K J cl, c is y rel umber L K, b d K E:. Fid the limit of { } {3 + (-) } if it eists. b. Fid the limit of { b } { } if it eists. E: Show tht the sequece whose th term is coverges Squeeze Theorem for Sequeces If lim L lim b d there eists iteger N such tht < c < b for ll > N, the lim c L Absolute Vlue Theorem For sequece { }, if lim the lim E: Show tht { c } ( ) coverges, d fid its limit.! Mootoic Sequeces d Bouded Sequeces Defiitio of Mootoic Sequece A sequece { } is mootoic if ll its terms re o-decresig or oicresig 3... or 3... E: Determie whether ech sequece is mootoic. 3 ( ) + b. b c. + c

3 Defiitio of Bouded Sequece. A sequece { } is bouded bove if there is rel umber M such tht M for ll. The umber M is clled upper boud of the sequece.. A sequece { } is bouded below if there is rel umber M such tht N for ll. The umber N is clled lower boud of the sequece. 3. A sequece { } is bouded if it is bouded bove d below. Bouded Mootoic Sequeces If sequece { } is bouded d mootoic, the it coverges. E:. The sequece { } { } is both bouded d mootoic d so, by the lst Thm must coverge. b. The diverget sequece { } { } b ( ) + is mootoic, but ot bouded. (its oly bouded below) c ( ) is bouded but ot mootoic. c. the diverget sequece { } { } Series d Covergece If { } is ifiite sequece the is ifiite series. Def of Coverget d Diverget Series For ifiite series, the th prtil sum is give by S If the sequece of prtil sums {S } coverges to S, the the series coverges. The limit S is clled the sum of the series. S If {S } diverges, the the series diverges.

4 E:. b. + c. The series i emple b is telescopig series of the form b b + b b + b b + b g b g b g... the th prtil sum is S b b + is follows tht telescopig series coverges iff b + pproches fiite umber s. If the series coverges the S b lim b +. E: 4 Geometric Series r + r+ r r +..., is geometric series with rtio r. Covergece of Geometric Series A geometric series with rtio r diverges if r. If < r <, the the the series coverges d r r E:. 3 b. 3 Properties of Ifiite Series If A, b B d c is y rel umber, the the followig series coverges to the idicted sums.. 3. c ca ( + b) A+ B ( b) A B

5 th Term Test for Divergece The followig thm sttes tht if series coverges, the limit of its th term must be. Limit of th Term of Coverget Series If coverges, the lim. th Term Test for Divergece If lim, the diverges. E:. b.! c.! + The Itegrl Test d p-series The Itegrl Test If f is positive, cotiuous, d decresig for d f(), the d ( ) f d either both coverge or both diverge. E:. b. + + p Series d Hrmoic Series p p p p is clled p series where p is positive 3 costt. If p, is clled Hrmoic series. 3 A geerl hrmoic series is of the form Σ/(+b).

6 Covergece of p - Series The p series p p p p 3. coverges if p >, d. diverges if < p <. E: Determie whether the followig series coverges or diverges: l Comprisos of Series Direct Compriso Test Let < b for ll.. If b. If coverges, the diverges, the b coverges. diverges. E: Determie the covergece or divergece of E: Determie the covergece or divergece of Limit Compriso Test Suppose tht >, b >, d lim L b where L is fiite d positive. The the two series Σ d Σb either both coverge of both diverge. E: Show tht the followig geerl hrmoic series diverges., >, b> + b

7 E: Determie the covergece or divergece of. + E: Determie the covergece or divergece of 3 4. Altertig Series + A series where the terms cotiuously switch from positive to egtive or vice vers is kow s ltertig series. E: ( ) Altertig Series Test Let >. The ltertig series ( + ) d ( ) coverge if the followig two coditios re met.. lim +,. for ll E: Determie the covergece or divergece of E: Determie the covergece or divergece of E: Does the Altertig Series Test pply to ( ) + ( ) + ( ) ( + ) Altertig Series Remider If coverget ltertig series stisfies the coditio +, the the bsolute vlue of the remider R N ivolved i pproimtig the sum S by S N is less th (or equl to) the first eglected term. Tht is, S SN RN N+

8 E: Approimte the sum of the followig series by its first si terms. + ( )! Absolute Covergece If the series coverges, the lso coverges. Def of Absolute Vlue d Coditiol Covergece. is bsolutely coverget if coverges.. is coditiolly coverget if coverges d diverges. E: Determie whether ech of the series is coverget or diverget. Clssify y coverget series s bsolutely or coditiolly coverget. ( )!. ( ) b. + c. b g ( )/ ( ) d. 3 l( + ) The Rtio d Root Tests Rtio Test Let Σ be series with ozero terms.. Σ coverges bsolutely if lim. Σ diverges if lim + > + < or lim + 3. The Rtio Test is icoclusive if lim + E: Determie the covergece or divergece of!

9 E: Determie whether ech series coverges or diverges. +. b. 3 c. ( ) Root Test Let Σ be series!. Σ coverges bsolutely if lim <.. Σ diverges if lim > or lim 3. The Root Test is icoclusive if lim + <.. E: Determie the covergece or divergece of e. Guidelies for Testig Series for Covergece or Divergece.. Does the th term pproch? If ot the series is diverget.. Is the series oe of the specil types: geometric, p-series, telescopig, or ltertig? 3. C the Itegrl Test, Root Test, or Rtio Test be pplied? 4. C the series be compred fvorbly to oe of the specil types? Determie the covergece or divergece of ech series. +. π b. 3+ F I HG 6 K J c. e d. 3 + e. 3 b g! f. 4+ g. + F + HG I K J

10 Power Series Def of Power Series If is vrible, the ifiite series of the form is clled power series. More geerlly, ifiite series of the form ( c) + ( c) + ( c) +... ( c) +... is clled power series cetered t c, where c is costt. E: Fid the ceter of the followig power series. b. ( )( ) + c. ( )! Rdius d Itervl of Covergece. A power series c be thought of s fuctio f where the domi of f is the set of ll vlues for which the power series coverges. The series lwys coverges t its ceter c, so c lwys lies i the domi of f. Covergece of Power Series For power series cetered t c, precisely oe of the followig is true.. The series coverges oly t c.. There eists rel umber R > such tht the series coverges bsolutely for c < R, d diverges for c > R. 3. The series coverges bsolutely for ll. The umber R is the rdius of covergece of the power series. If the series coverges oly t c, the rdius of covergece is R If the series coverges for ll, the rdius of covergece if R. The set of ll vlues of for which the power series coverges is the itervl of covergece of the power series.

11 E: Fid the rdius of covergece of E: Fid the rdius of covergece of! 3( ) + ( ) E: Fid the rdius of covergece of (+ )! Edpoit Covergece Whe r is fiite umber the previous thm sys othig bout the covergece t the edpoits of the itervl (there is o or equl to) so they eed to be tested seprtely. E: Fid the itervl of covergece of E: Fid the itervl of covergece of E: Fid the itervl of covergece of ( ) ( + )

12 Differetitio d Itegrtio of Power Series. Power series represettio of fuctios hs plyed importt role i Clculus. Much o Newtos work with differetitio itegrtio ws doe i the cotet of power series. Properties of Fuctios Defied by Power Series If the fuctio give by f( ) ( c) + ( c) + ( c) +... hs rdius of covergece R >, the, o the itervl (c R, c + R), f is differetible (d therefore cotiuous). Moreover, the derivtive d tiderivtive of f re s follows... f ( ) ( c) + c + c + ( ) 3 3( )... f( ) d C + ( c) + + ( c) C+ ( c) The rdius of covergece of the series obtied by differetitig or itegrtig power series if the sme s tht of the origil power series. The itervl of covergece, however, my be differet s result of the behvior t the edpoits. 3 E: For f( ) Fid z the itervls of covergece for ech of the followig. f( ) d b. f () c. f ( )

13 Represettio of Fuctios by Power Series Geometric Power Series: I this sectio we will look t represetig fuctio by power series. Cosider f( ) this closely resembles the geometric series r, r < if d r therefore r E: Fid the power series for E: Fid the power series for 4 f( ) +, cetered t f( ) Opertios with Power Series Let f( ) d g( ) b. f( k) k N N. f( ) 3. f( ) ± g( ) ( ± b ), cetered t. 3 E: Fid the power series cetered t, for f( ) Tylor d Mcluri Series The Form of Coverget Power Series If f is represeted by power series f () Σ ( c) for ll i ope itervl I cotiig c, the f () (c)/! d ( ) f ( c) f ( c) f( ) f( c) + f ( c)( c) + ( c) ( c) +...!!

14 Def of Tylor Series d Mcluri Series If fuctio f hs derivtives of ll orders t c, the the series ( ) f ( c) f ( c) ( c) f( c) + f ( c)( c) + ( c) +...!! is clled the Tylor series for f () t c. Moreover, if c, the the series is the Mcluri series for f. If you kow the ptter for the coefficiets of the Tylor polyomil for fuctio, you c eted the ptter esily to form the correspodig Tylor series. E: Use the fuctio f () si to form the Mcluri series d determie the itervl of covergece. Covergece of Tylor Series If lim for ll i the itervl I, the the Tylor series for f R coverges d equls f (). ( ) f ( c) f( ) ( c)! f ()! Guidelies for Fidig Tylor Series. Differetite f () severl times d evlute ech derivtive t c. Try to recogize ptter i these umbers.. Use the sequece developed i the first step to form the Tylor coefficiets f () (c)/! d determie the itervl of covergece for the resultig power series. 3. Withi the itervl of covergece, determie whether or ot the series coverges to f (). E: Fid the Mcluri series for f () si ( )