Sequeces A Sequece is a list of umbers writte i order. {a, a 2, a 3,... } The sequece may be ifiite. The th term of the sequece is the th umber o the list. O the list above a = st term, a 2 = 2 d term, a 3 = 3 rd term, etc... Example I the sequece {, 2, 3, 4, 5, 6,... }, we have a =, a 2 = 2,.... The th term is give by a =. Some sequeces have patters, some do ot. Example If I roll a 20 sided die repeatedly, I geerate a sequece of umbers, which have o patter. Example The sequeces {, 2, 3, 4, 5, 6,... } ad have patters. {,,,,,... } Aette Pilkigto Lecture 23 : Sequeces
Formula for a Sometimes we ca give a formula for the th term of a sequece, a = f (). Example For the sequece {, 2, 3, 4, 5, 6,... }, we ca give a formula for the th term. a =. Example Assumig the followig sequeces follow the patter show, give a formula for the -th term: {,,,,,... } th term = a = ( ) +. { /2, /3, /4, /5, /6,... } th term = a = ( ) +. Factorials are commoly used i sequeces 0! =,! =, 2! = 2, 3! = 3 2,...,! = ( ) ( 2). Example Fid a formula for the th term i the followig sequece ( ) 2, 4 2, 8 6, 6 24, 32 20,..., a =, th term = a = 2!. Aette Pilkigto Lecture 23 : Sequeces
Differet ways to represet a sequece Below we show 3 differet ways to represet the sequeces give: A. 2, 2 3, 3 4,..., +, B. C. 3... o, + = 3, 5 9, 7 27,..., ( ) (2 + ) 3,..., a = +. o, ( ) (2 + ), a 3 = ( ) (2 + ). = 3 e, e 2 2, e 3 6,..., e o!,..., e! =, a = e!. Aette Pilkigto Lecture 23 : Sequeces
Graph of a sequece A sequece is a fuctio from the positive itegers to the real umbers, with f () = a. We ca draw a graph of this fuctio as a set of poits i the plae. The poits o the graph are (, a ), (2, a 2), (3, a 3),..., (, a ),... Example Graph the sequeces { ( ) } = ad 2 3 3 =. 0.0 2.0000 0.05.9999.9998 20 40 60 80 0.05.9997 0.0.9996 20 40 60 80 00 We ca see from these pictures that the graphs get closer to a horizotal asymptote as, y = 0 o the left ad y = 2 o the right. Algebraically this meas that as, we have ( ) 0 ad 23 2. 3 Aette Pilkigto Lecture 23 : Sequeces
Limit of a sequece Defiitio A sequece {a } has limit L if we ca make the terms a as close as we like to L by takig sufficietly large. We deote this by lim a = L or a L as. If lim a exists (is fiite), we say the sequece coverges or is coverget. Otherwise, we say the sequece diverges. Graphically: If lim a = L, the graph of the sequece {a } = has a uique horizotal asymptote y = L. Equivalet Defiitio A sequece {a } has limit L ad we write lim a = L or a L as if for every ɛ > 0 there is ad iteger N with the property that if > N the a L < ɛ. Aette Pilkigto Lecture 23 : Sequeces
Determiig if a sequece is coverget. Usig our previous kowledge of limits : Theorem If lim x f (x) = L ad f () = a, where is a iteger, the lim a = L. Example Determie if the followig sequeces coverge or diverge: A. 2, B. 2 = 2 3 3 = 2 A. lim x 2 x = lim x 2 x x =. o 2 Therefore the sequece coverges ad 2 = 2 lim a = lim =. 2 2x B. lim 3 2 /x x = lim 3 x 3 x = 2. o 2 Therefore the sequece 3 coverges ad 3 = 2 lim a = lim 3 = 2. 3 Aette Pilkigto Lecture 23 : Sequeces
L Hospital s rule We ca use L Hospital s rule to determie the limit of f (x) if we have a idetermiate form. Example Is the followig sequece coverget? 2 = lim x x = (by l Hospital) lim 2 x x = 0. 2 x l 2 Therefore the sequece coverges ad im 2 = 0. Divergig t. lim a = meas that for every positive umber M, there is a iteger N with the property if > N, the a > M. I this case we say the sequece {a } diverges to ifiity. Note: If lim x f (x) = ad f () = a, where is a iteger, the lim a =. Aette Pilkigto Lecture 23 : Sequeces
Importat sequece/limit Example Show that the sequece {r } =, r 0, coverges if 0 r ad diverges to ifiity if r >. lim r = lim x r x = lim x e x l r. 8 < 0 if r < lim x e x l r = if r = : if r > Therefore the sequece {r } =, r 0, coverges if 0 r ad diverges to ifiity if r >. Aette Pilkigto Lecture 23 : Sequeces
Rules of Limits The usual Rules of Limits apply: If {a } ad {b } are coverget sequeces ad c is ay costat the lim (a + b) = lim a + lim b lim (a b) = lim a lim b lim c = c lim ca = c lim a lim (ab) = lim a lim b a lim a lim = b lim b h i p lim ap = lim a if p > 0 ad a > 0 I fact if lim a = L ad f (x) is a cotiuous fuctio at L, the lim f (a) = f (L). if lim b 0 Aette Pilkigto Lecture 23 : Sequeces
Applyig the Rules of Limits Example Determie if the followig sequece coverges or diverges ad if it coverges fid the limit. r 3 2 +. = q q lim ( 3 2+ ) = lim 3 2+ lim = 3 qlim 2+ lim = 3 qlim x 2x+ x lim x x = 3 q lim x 2+/x 0 = 3 2 q 3 Therefore the sequece 2+ = coverges to 3 2. Aette Pilkigto Lecture 23 : Sequeces
Whe there is o f(x) / Squeeze Theorem Note We caot always fid a fuctio f (x) with f () = a. The Squeeze Theorem or Sadwich Theorem ca also be applied : If a b c for 0 ad lim a = lim c = L, the lim b = L. Example Fid the limit of the followig sequece 2! =, Requires a bit of cleveress, because we caot replace! by a fuctio x!. Certaily 2 > 0 for all. So if we ca fid a sequece {c! } with 2 c! for all ad lim c = 0, the we ca apply the squeeze theorem. Note that 2 = 2 2 2 2 2! 2 3 Sice 2 2 if k 2, we have 2 2 for all 2. k! Sice lim 2 2 2 = 0, ad 0 2 2 for all 2, we ca coclude! 2 that lim = 0 usig the squeeze theorem.! Therefore the sequece 2! = Aette Pilkigto coverges to 0. Lecture 23 : Sequeces
Alteratig Sequeces Theorem If {a } is a alteratig sequece of the form ( ) a where a > 0, the the alteratig sequece coverges if ad oly if lim a = 0 or (for the sequece described above) lim a 0. (also true for sequeces of form ( ) + a or ay sequece with ifiitely may positive ad egative terms) Example Determie if the followig sequeces coverge: A. ( ) 2 +, B. ( ) 2 + 2 = = A. a = ( ) 2+ 2. 2+ 2x+ (2/x)+(/x lim a = lim = lim 2 x = lim 2 ) x 2 x = 0 Therefore the sequece ( ) 2+ coverges to 0. 2 = B. b = ( ) 2+. 2+ 2x+ lim b = lim = lim x x Therefore the sequece diverges. ( ) 2+ = = lim x 2+(/x) = 2 0. Aette Pilkigto Lecture 23 : Sequeces
Alteratig Sequeces Geometrically, we ca see the differece i the behavior of the sequeces above by examiig their graphs. The coverget sequece has a uique horizotal asymptote whereas the diverget sequece has two. 0.2 2 0. 20 40 60 80 20 40 60 80 0. 2 0.2 3 Aette Pilkigto Lecture 23 : Sequeces
Mootoe Bouded Sequeces Defiitio A sequece {a } is called icreasig if a < a + for all, or a < a 2 < a 3 <.... A sequece {a } is called decreasig if a > a + for all, or a > a 2 > a 3 >.... A sequece {a } is called mootoic if it is either icreasig or decreasig. Defiitio A sequece {a } is bouded above if there is a umber M for which a M for all. A sequece {a } is bouded below if there is a umber m for which a m for all. A sequece that is bouded above ad below is called Bouded. Theorem Every bouded mootoic sequece is coverget. (This theorem will be very useful later i determiig if series are coverget.) Aette Pilkigto Lecture 23 : Sequeces
Mootoe Bouded Sequeces, Example To check for mootoicity If we have a differetiable fuctio f (x) with f () = a, the the sequece {a } is icreasig if f (x) > o ad the sequece {a } is decreasig if f (x) < o. Example Show that the followig sequece is mootoe ad bouded ad hece coverges. {ta ()} = We kow that π < 2 ta () < π for all > 0. 2 We also kow that ta () icreases as icreases, sice d ta x = > 0 for all x. dx x 2 + Therefore, we ca coclude that the sequece above coverges. We could actually compute the limit here, but usig the theorem for bouded mootoic sequeces, we have cocluded that the sequece coverges without directly computig the limit. Aette Pilkigto Lecture 23 : Sequeces