Ifiite sequeces (Sect. 0. Today s Lecture: Review: Ifiite sequeces. The Cotiuous Fuctio Theorem for sequeces. Usig L Hôpital s rule o sequeces. Table of useful its. Bouded ad mootoic sequeces. Previous Lecture: Overview: Sequeces, series, ad calculus. Defiitio ad geometrical represetatios. The it of a sequece, covergece, divergece. Properties of sequece its. The Sadwich Theorem for sequeces. Review: Ifiite sequeces Defiitio A ifiite sequece of umbers is a ordered set of real umbers. Defiitio A ifiite sequece {a } has it L iff for every umber ɛ > 0 there exists a positive iteger N such that N < a L < ɛ. A sequece is called coverget iff it has a it, otherwise it is called diverget. Remark: The its of simple sequeces ca be used to compute its of more complicated sequeces.
Ifiite sequeces (Sect. 0. Review: Ifiite sequeces. The Cotiuous Fuctio Theorem for sequeces. Usig L Hôpital s rule o sequeces. Table of useful its. Bouded ad mootoic sequeces. The Cotiuous Fuctio Theorem for sequeces Theorem If a sequece {a } L ad a cotiuous fuctio f is defied both at L ad every a, the the sequece {f (a } f (L. { ( (2 + + 3 2 } Fid the it of l (2 2 as. + 3 ( (2 + + 3 2 Solutio: The sequece b = l (2 2 ca be writte as + 3 b = f (a, f (x = l(x, a = (2 + + 32 (2 2. + 3 a = (2 + + ( 32 ( 2 + 2 (2 2 ( + 3 = 2 ( + 3 3 2 + 3 2. 2 ( 2 3 We coclude that b l. 2
Ifiite sequeces (Sect. 0. Review: Ifiite sequeces. The Cotiuous Fuctio Theorem for sequeces. Usig L Hôpital s rule o sequeces. Table of useful its. Bouded ad mootoic sequeces. Usig L Hôpital s rule o sequeces Theorem (L Hôpital s rule for sequeces If the sequece {a } satisfies that: There exist a fuctio f such that for > N the sequece elemets a ca be writte as a = f (; Ad f (x = L; the holds that a = L. Remark: The might be used to compute that it. Fid the it a = 8 5 as. f (x may idetermiate, ad L Hôpital s rule Solutio: Notice that a = f ( for f (x = 8x 5x.
Usig L Hôpital s rule o sequeces Fid the it a = 8 5 as. Solutio: Recall: a = f ( for f (x = 8x 5x. 8x 5x = e l( 8x 5x l(5x = e( 8x But l(5x 8x is idetermiate. L Hôpital s rule, l(5x 8x = ( 8x 5x = e l(5x 8x ( x 8 = 8x = 0. = e 0 8x 5x =. We coclude that 8 5 as. Usig L Hôpital s rule o sequeces ( Give positive umbers a, b, fid the b a. Solutio: We rewrite the sequece as follows, ( b a ] = e [a l( b = e [ a l( b ] The expoet has a idetermiate it, a l( b 0 0. Recall the argumet with the L Hôpital s rule o fuctios, a l( b x x We coclude that = a b ( b x x 2 x 2 = ab ( b x = ab. ( b a = e ab.
Ifiite sequeces (Sect. 0. Review: Ifiite sequeces. The Cotiuous Fuctio Theorem for sequeces. Usig L Hôpital s rule o sequeces. Table of useful its. Bouded ad mootoic sequeces. Table of useful its Remark: The followig its appear ofte i applicatios: l( x ( = 0; = ; =, for x > 0; x = 0, for x < ; ( x + = e x, for x R; x! = 0.
Ifiite sequeces (Sect. 0. Review: Ifiite sequeces. The Cotiuous Fuctio Theorem for sequeces. Usig L Hôpital s rule o sequeces. Table of useful its. Bouded ad mootoic sequeces. Bouded ad mootoic sequeces Defiitio A sequece {a } is bouded above iff there is M R such that a M for all. The sequece {a } is bouded below iff there is m R such that m a for all. A sequece is bouded iff it is bouded above ad below. a = is bouded, sice 0 <. a = ( is bouded, sice (.
Bouded ad mootoic sequeces Defiitio A sequece {a } is icreasig iff a < a +. A sequece {a } is o-decreasig iff a a +. A sequece {a } is decreasig iff a > a +. A sequece {a } is o-icreasig iff a a +. A sequece is mootoic iff the sequece is both o-icreasig ad o-decreasig. Theorem A o-decreasig, bouded above sequece, coverges. A o-icreasig, bouded below sequece, coverges. Bouded ad mootoic sequeces Determie whether the sequece a = 2 + coverges or ot. Solutio: We show that a is decreasig. Ideed, the coditio a + < a + ( + 2 + < 2 + ( + ( 2 + < ( 2 + 2 + 2 3 + 2 + + < 3 + 2 2 + 2 Sice < ( 2 + is true for, the a + < a ; decreasig. The sequece satisfies that 0 < a, bouded below. We coclude that a coverges.