Infinite Sequences and Series. Sequences. Sequences { } { } A sequence is a list of number in a definite order: a 1, a 2, a 3,, a n, or {a n } or

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1 Mth 0 Clculus II Ifiite Sequeces d Series -- Chpter Ifiite Sequeces d Series Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. Sequeces Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. Sequeces A sequece is list of umber i defiite order: or { } or แ Exmple: } { { } { } cos cos 6 cos d) 0 c) ) ) ) ) ) ) b) 5 ) 0 π π π

2 Exmple pge 78 Fid formul for the geerl term of the sequece แ Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. Problem pge 77 Fid formul for the geerl term of the sequece แ Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. 5 Fibocci Sequece Fid formul for the geerl term of the sequece แ Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. 6

3 Defiitio pge 79 A sequece { } hs the limit L lim L or L s If lim exists we sy the sequece coverges or is coverget. Otherwise the sequece diverges or is diverget. Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. 7 Defiitio pge 79 A sequece { } hs the limit L lim L or L s If for every ε > 0 there is correspodig iteger N such tht L < ε wheever > N Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. 8 Theorem pge 70 If lim f x) L d f ) x whe is iteger the lim Exmple : lim 0 for r > 0 sice lim 0 for r > 0 x r x r L. Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. 9

4 Defiitio 5 pge 70 lim mes tht for every positive umber M There is iterger N such tht > M wheever > N. Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. 0 Limit Lws for Sequeces If { }d{ b}re coverget sequeces d c is costt the lim + b ) lim + lim b lim b ) lim lim b lim c c lim lim c c lim b ) lim lim b lim lim lim b 0 b lim b lim lim p p if p > 0 d > 0 Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. Theorem 6 pge 7 If lim 0 the lim 0 ) Exmple7: Evlute lim if it exists. Exmple6 : Determieif the sequece ) is coverget or diverget. Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter.

5 Squeeze Theorem pge 7 If b c for 0 d lim lim c L the lim b L! Exmple8 : Determie if the sequece where! L. is coverget or diverget. Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. Exmple 9 pge 7 For wht vlues of r is the sequece if r > lim r if r 0 if 0 < r < { r } coverget? Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. Defiitio 9 pge 7 A sequece { } is clled icresig if < + for ll tht is < < <. A sequece { } is clled decresig if > + for ll tht is > > >. It is either mootoiclly icresig or decresig. Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. 5 5

6 Defiitio 0 pge 7 A sequece { } is bouded bove if there is umber M such tht M for ll A sequece { } is bouded below if there is umber m such tht m for ll If it is bouded bove d below the { } is bouded sequece. Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. 6 Mootoic Sequece Theorem Every bouded mootoic sequece is coverget. Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter Problems o Pge 77 ) cos / ) { rct } 6. Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. 8 6

7 Prctice Problems o Pge odds Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. 9 Series Mth 0 Clculus II Ifiite Sequeces d Series: Series -- Chpter. 0 Give series If If the sequece the the series Defiitio pge 750 { s } + + L let s is coverget d lim s i i + + L+ + L s or The umber s is clled the sum of the series. Otherwise the series is clled diverget. s + + L+ is clled coverget d we write deote its th prtilsum : s exists s rel umber s Mth 0 Clculus II Ifiite Sequeces d Series: Series -- Chpter. 7

8 The Geometric Series The geometricseries is coverget if r < d its sum is + r + r r r < r If r the geometricseries is diverget. r + L Mth 0 Clculus II Ifiite Sequeces d Series: Series -- Chpter. 0 ). Problems o Pge 756 Determie whether theseriesis coverget or diverget. If it is coverget fid its sum L 8 6) Mth 0 Clculus II Ifiite Sequeces d Series: Series -- Chpter. If the series Exmple: 6. Theorem 6 pge 75 l + is coverget the lim 0. Mth 0 Clculus II Ifiite Sequeces d Series: Series -- Chpter. 8

9 The Test for Divergece If lim. 0. does ot exist or if lim If it is coverget fid its sum. + ) + ) l the series Determie whether the series is coverget or diverget. is diverget. Mth 0 Clculus II Ifiite Sequeces d Series: Series -- Chpter. 5 If d Theorem 8 pge 755 b re coverget series d c is costt the c is coverget. + b ) is coverget. b ) is coverget. c c + b ) + b ) b b Mth 0 Clculus II Ifiite Sequeces d Series: Series -- Chpter. 6. Problems o Pge 756 Determie whether the series is coverget or diverget. If it is coverget fid its sum Fid the vluesof x for which the series coverges Fid the sum of the series for those vluesof x + ) x. Mth 0 Clculus II Ifiite Sequeces d Series: Series -- Chpter. 7 9

10 Prctice Problems o Pge odds 5 Mth 0 Clculus II Ifiite Sequeces d Series: Series -- Chpter. 8 0