Natioal Jio College Mathematics Depatmet 00 Natioal Jio College 00 H Mathematics (Seio High ) Seqeces ad Seies (Lecte Notes) Topic : Seqeces ad Seies Objectives: At the ed of this topic, stdets shold be able to destad the meaig of fiite ad ifiite seqeces ad seies, ad se the otatio to deote the th tem of a seqece o seies kow that a seqece ca be geeated by a fomla fo the th tem, o by a ecece elatio se otatio to expess the sm to tems ( S ) of a seies se the method of diffeeces to obtai the sm to tems of a seies give the geeal tem destad that a ifiite seqece o seies ca covege, ad fid the limit of the seqece o the sm to ifiity of a coveget seies Itodctio The followig diagam is a oveview of what we ll be coveig fo this topic Seqeces Seies Repesetatio Limit Fomla fo the tem th Recece Relatio Rles fo Smmatio Method of Diffeece Covegece 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio)
Natioal Jio College Mathematics Depatmet 00 Defiitio of a seqece A seqece is a set of tems o mbes i a defied ode It is ofte witte as:,,, 4,,, whee is the th tem (o geeal tem) Example Coside the followig sets of mbes: (i), 4, 6, 8, 0,, 4 (iii) 4 5 6,,,, 4 6 8 0 (ii),, 9, 8, 4, (iv),,,, 5, 8,, (i) to (iv) ae examples of mbe seqeces Each sbseqet tem i the seqece ca be defied by a fomla o le Qestio: Ca yo defie the above mbe seqeces by a fomla o le? Soltios: (i),,,, 7 (ii) (iii) + ( + ) (iv) +,, whee (Icidetally, this seqece is called the Fiboacci s Seqece) whee epesets the tem mbe ad epesets the th tem Seqeces ca be geeated by a fomla fo the th tem, o by a ecece elatio Fiite ad Ifiite Seqece A fiite seqece is a seqece whee the mbe of tems is fiite, ie it will always have a last tem A ifiite seqece is a seqece whee the mbe of tems is ifiite, ie it does ot have a edig tem Qestio: Ca yo idetify the fiite ad ifiite seqeces i Example? Fiite seqece: (i) Ifiite seqeces: (ii) (iv) 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio)
Natioal Jio College Mathematics Depatmet 00 Seqeces geeated by a fomla fo the th tem To defie sch a seqece with the mbes i a give ode ad a obvios le fo obtaiig each sbseqet tem, we sally eqie types of ifomatio: (i) st tem,, (ii) a le by which the geeal tem ca be calclated, (iii) the mbe of tems, (Uless it is a ifiite seqece) Example Wite dow the fist fo tems of the seqece i which the geeal tem,, is + Soltio: 4 + + 5 + 0 + 4 7 Example Wite dow the le fo the geeal tem of the followig seqece:, 5, 8,, 4, Soltio: Seqeces geeated by a ecece elatio A ecece seqece is a seqece of mbes whee each tem i the seqece is obtaied fom oe o moe of the pevios tems To defie sch a seqece, we sally eqie types of ifomatio: Example 4 (i) st tem (o the fist few tems), (ii) a ecece elatio of the fom + f( ) [ca elate to moe pevios tems Bt eqied by the syllabs to cove p to ode oly] A seqece,,, 4,, is defied by ad + fo > 0 Wite dow the fist 4 tems of the seqece Soltio: () () 5 (5) 4 4 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio)
Natioal Jio College Mathematics Depatmet 00 Example 5 The seqece of mbes,,, 4,, is defied by 6, 0 ad + 6+ 8 fo > Fid ad 5 Soltio: 6 8 6(0) 8(6) 7 6 8 6(7) 8(0) 7 4 6 8 6(7) 8(7) 056 5 4 Ca yo see that the seqece of Fiboacci mbes is a example of a ecece seqece? 4 Limit of a Seqece Coside a ifiite seqece whee the th tem is defied as, What ca yo say abot the th tem of the seqece whe teds to ifiity ie takes o vey lage vales?,,,,,,,, 4 00 500 Obseve that the seqece of mbes get close to 0 (bt eve eqals 0) as gets lage Ths, we say the seqece coveges to 0 as teds to ifiity We wite: As, 0 o lim 0 A ifiite seqece,,, 4, has a limit L if ad oly if the tems get close to L (bt ot eqal to L) as gets lage ad lage We say that the seqece is coveget ad it coveges to the vale of L Othewise the seqece is diveget if it does ot have a limit Example 6 Detemie if the followig seqeces (give the geeal th tem) ae coveget o diveget Fid the limit of the seqece if it coveges (a) ( ), (b) +, (c) 4 + Soltio: (a) seqece:,,,,, we see that the seqece alteates betwee ad, ad hece the seqece is diveget 4 4 0 (b) lim lim (c) lim lim 0 + + + 0 + + + 0 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 4
Natioal Jio College Mathematics Depatmet 00 Example 7 A seqece of positive mbes is give ecsively by the elatio 7a + 9 a + fo,, It is kow that as, a l Fid the exact vale of l Soltio: As, a l ad a + l 7a + 9 7l + 9 a+ l l l 7 9 0 Refe to Sectio fo the soltio sig GC Solvig the eqatio gives l 45 o l (NA as l > 0) Qestio: Why mst l > 0? Seies ad Σ Notatio Defiitio of a Seies A seies is fomed whe the tems of a seqece ae added ad it ca eithe be fiite o ifiite The seies fomed fom the seqece,,,,, is + + + + + Qestio : What is the diffeece betwee a seqece ad a seies? Example (i) + + ad + 5 + 0 + 7 + + ( + ) ae fiite seies (ii) + + + ad + + + 4 + + ae ifiite seies 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 5
Natioal Jio College Mathematics Depatmet 00 The Sigma Notatio Deotes the highest vale that takes i the smmatio Geeal tem of the seqece Deotes the lowest vale that takes i the smmatio is teated as a dmmy vaiable i the smmatio It is commoly eplaced by othe alphabets sch as i, j, k etc Illstatio: 0 epesets the seies + + + 0 It also meas the sm of all tems of the fom whee takes all itegal vales fom to 0 iclsive Qestio : How do we epeset the seies (i) & (ii) i Example i smmatio otatio? Soltios: (i) + + & + 5+ 0 + 7 + + ( + ) ( + ) (ii) + + + lim & + + + + + lim Impotat Notes: S deotes the sm of the fist tems of a seies, ie S + + + + S deotes the sm to ifiity of a seies, ie S + + + + + S S b 4 a + a+ + a+ + a+ + + b + b, a < b ε Z a 5 Total mbe of tems i b a + b a 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 6
Natioal Jio College Mathematics Depatmet 00 Example (a) Wite ot the followig seies explicitly 5 (i) ( ) (() ) + (() ) +(() ) +((4) ) +((5) ) 5 (ii) i + + + 4 + 5 i 4 (iii) ( ) ( ) + ( ) +(4 ) (iv) ( ) 5 j j j ( ) +( ) + ( ) + ( ) 4 4 + ( ) 5 5 (b) Wite dow each of the followig seies i smmatio otatio (i) 0! +! +! +! + + 6! 6 k 0 k! (ii) k + x + x + x + + x k 0 x (iii) + + 5 + 7 + +( ) to tems ( k ) o ( k + ) k k 0 (v) + + 5 + to tems 4 ( ) ( + ) ( ) m 4 8 6 m ( ) (vii) + + + + ( ) m (iv) + 4 + 45 + 456 + to tems k ( k ) k + ( k + ) (vi) + 6 + 8 + 54 + to tems (viii) k () o k 0 k () k a + a a + ( ) ( ) ( ) ( ) ( ) a 0 Note: The same sm ca be expessed i diffeet ways Eg ( ) ( + ) 0 The cote is a dmmy vaiable Eg ( ) ( + i + ) ( k + ) i k If the seies cotais ifiitely may tems, we eplace by Eg + + 4 4 To expess a seies i smmatio otatio, it is ecessay to fid the geeal tem 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 7
Natioal Jio College Mathematics Depatmet 00 Rles fo Smmatio Sm of a costat: If a is a costat idepedet of, the a a + a + a + a + + a + a + a a times Note: k k ad k (k +) 0 Diffeece of Sms Show that m m LHS m + m+ + m+ + + - + ( + + + m- + m + m+ + + - + ) ( + + + m- ) RHS Distibtive Popety of Sms: If a, b ad c ae costats, idepedet of, the ( a + b) a + b m m m ( ) ( ) ( ) ( ) ( ) af + bg( ) + ch + a f + b g + c h + 4 Natal Nmbe Seies (Some Special Seies) (a) + + + 4 + + ( + ) (Sm of cosective whole mbes) (b) + + + + ( + )( + ) (Sm of Sqaes) 6 Note: (c) ( ) ( ) + + + + 4 + + ie f ( ) g( ) f ( ) g ( ) (Sm of Cbes) 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 8
Natioal Jio College Mathematics Depatmet 00 Example Evalate (a) ( + ) + 6 + + ) ( ( )( ) ( ( ) + ) + [(+)(+) (+) +] (b) 0 ( )( + ) 0 ( ) + 0 0 0 0 + + ( ) 4 6 970 (0) () (0)()(4) 0 () 0 4 Smmatio by the Method of Diffeeces If the geeal tem,, of a seies ca be expessed as a diffeece of two fctios, witte as f( + ) f(), the [ f ( + ) f ( ) ] f ( + ) f () + f ( + ) f () + f ( + ) f () + + f (( ) + ) f ( ) + f ( + ) f ( ) f ( + ) f () Note that simila tems i the smmatio ae cacelled This is kow as the method of diffeeces 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 9
Natioal Jio College Mathematics Depatmet 00 Example 4 Use the idetity ( + ) + + to show that Soltio: ( + ) ( + ) 6 Hece, Example 5 Smmig both sides of the idetity, [( + ) ] [ + + ] + + Note that the LHS ca be simplified as Theefoe, we have [( + ) ] [( + ) ] + [( ) ( ) ] + [( ) ( ) ] + [(4) ] + [() ] + ( + ) + + [() ] ( + ) ( + ) ( + ) + + + ( + ) ( + ) (Show) 6 ( ) ( )( ) Show that ( + )! ( )!!( + ) Hece fid Soltio: + +!( ) +!( ) ( + )! ( + )! ( + )! ( )! [ ] (!) (!) ( + )! ( )! ( + )! ( )! + (4!) (!) + 4(5!) (4!) + + ( + )! ( )(!) ( + )! 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 0
Natioal Jio College Mathematics Depatmet 00 5 Covegece of a Seies A fiite seies has a fiite sm Howeve a ifiite seies may o may ot have a fiite sm A seies havig a fiite sm is said to be a coveget seies Fo example, (a) (b) The fiite seies 0 has a fiite sm of 55, ie 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 0 + + + + 0 55 The ifiite seies has a fiite sm,, 4 ie + + + + (Will be discssed fthe i the topic Geometic Pogessio ) ** Teache ca se a sqae to illstate that the above-metioed sm to ifiity teds to (c) The ifiite seies (ad is kow as a diveget seies) does ot have a fiite sm Ths the seies does ot covege Example 6 Expess i patial factios Hece fid x ( x + ) x x( x + ) Soltio: A B Let + By cove - p le, A, B x ( x + ) x x + Coside, x x ( x + ) x x ( x + ) x x x + lim ( ) x x x + x x( x + ) + 4 ( + 5) lim 4( + )( + ) + 5 5 + lim 8 + 4 + + + + + 4 + + + ( + 5) 4( + )( + )
Natioal Jio College Mathematics Depatmet 00 Use of Gaphic Calclato i Seqece ad Seies (Self-Exploatio) Geeatig ad Smmig a Seqece Example (i) Wite dow the fist fo tems of the seqece i which the geeal tem,, is + (ii) Fid the sm to the fist fo tems of the seqece i (i) Method : Use of the opeato seq( Steps Pess OPS Pess ad select optio 5: seq(, de Sceeshot Ipt the 5 paametes i the followig ode, each sepaated by a comma: ) the fomla fo the geeal th tem ) the vaiable sed i the above fomla (X i this case) ) the fist vale that X takes 4) the last vale that X takes 5) icemet of X i the seqece Pess ad the GC will geeate the seqece of mbes accodig to the ipt give i Step 4 To fid the sm of the seqece, pess ad select optio 5:sm(, de MATH Pess ad The sm of the seqece will be compted 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio)
Natioal Jio College Mathematics Depatmet 00 Method : Steps Pess Ceatig a list of mbes de a table ad select optio : Edit de EDIT Sceeshot Pess Scoll p to select the headig L (it will be highlighted) Pefom Steps to as i Method to geeate a list of the tems of the seqece L() gives the st tem of the seqece To get the th tem, scoll dow the colm til yo each L() 4 To fid the sm of the seqece, pess to go to Home Scee pess optio 5:sm(, de MATH ad select Pess to select the list L ad The sm of the seqece will be compted 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio)
Natioal Jio College Mathematics Depatmet 00 Geeatig a Seqece by a Recece Relatio Example A seqece,,, 4,, is defied by ad + fo > 0 Wite dow the fist 4 tems of the seqece Steps Pess Sceeshot Seqece Mode to highlight SEQ Yo ae ow i Pess to access the Fctio Edito Me Ete miimm vale fo which is Ete the fomla fo as Ete fo (Mi) Pess to go to Home Scee To obtai the vales of,, ad 4, key i The fist 4 tems of the seqece is ow geeated 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 4
Natioal Jio College Mathematics Depatmet 00 GC Soltio to Example 7 A seqece of positive mbes a is give ecsively by the elatio 7a + 9 a + fo,, It is kow that as a l Fid the exact vale of l, Steps Pess to access the Fctio Edito Me Ete miimm vale fo which is Ete the fomla fo Sceeshot as Ete fo (Mi) Pess vales to set p the table of seqece Ete TblStat ad Tbl To view a table of seqece vales, pess By scollig dow the table, we obseve that the tems of the seqece () coveges to 45 Hece l 45 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 5
Natioal Jio College Mathematics Depatmet 00 Natioal Jio College 00 H Mathematics (Seio High ) Seqeces ad Seies (Ttoial) Basic Mastey Qestios Evalate the followig seies: 8 (a) ( + ) (b) ( 5 + ) (c) ( i + )( i + ) 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 6 0 i (leave yo aswe i tems of ) I the seqece,,, 4,,,, whee 8 ad + 4, fid 5 5 ad Wite dow the geeal th tem of the followig seies ad expess each seies i smmatio otatio: (a) + 4 + 7 + 0 + to 0 tems (b) + + 5 + 7 + to 0 tems (c) ( )( 4) + ( 4)( 5) + ( 5)( 6) + to tems (d) + + to tems 5 7 4 Expess the followig seies i the fom f ( ), oe begiig at 0 ad the othe begiig at : (a) ( )( ) ( )( 4) + ( )( 5) + + ( 0)( ) b a + (b) 00 5 Use the method of diffeece to evalate ( ) Ttoial Qestios 4 + 5 + 6 + + Evalate the followig seies, simplifyig yo aswe as fa as possible: m (a) ( m i) (b) ( + )( j + ) i 0 j Evalate the followig seies sig yo GC: 8 0 (a) ( ) (b) ( 0) j Give that i j (c) log a ( a ) j ( + )( ) + 6, fid (i) + + + + ( ) (ii) + 4 + 6 + + ( ) Hece, fid ( ) + + 5 + + i
Natioal Jio College Mathematics Depatmet 00 4 By expessig ( )( + ) i patial factios, evalate ( )( + ) Hece, fid (i) 5 (ii) ( )( + ) ( )( + ) 5 Expess ( )( + )( + ) that ( )( )( ) i patial factios Hece, o othewise, show ( + ), ad evalate the ifiite seies + + ( + )( + ) + + + 5 5 7 5 7 9 a 6 The seqece of eal mbes a0, a, a, is sch that a + + a method of diffeece, o othewise, fid the vale of a a 00 0 By sig the 7 Give that f( ) f ( ) f ( + ), whee is a positive itege, fid a sigle expessio fo Hece, fid the sm to tems of the seies + + + + 6 0 Dedce that the sm to tems of the seies + + + + is less tha 4 5 8 The th tem of a seqece is give by! ( ) Show that ( )!( ) + ad N + + ( ) + N + 5, fo all positive iteges whee ( N )! N! Hece fid the vale of N sch that ( ) ( )! 5 0 + 0 695640 9 The positive mbes x satisfy the elatio x ( ) + x + 5, fo,,, As, x l (i) Fid (i eithe ode) the vale of l to decimal places ad the exact vale of l (ii) Pove that ( ) (iii) x l x l Hece show that if x + > l, the x > x+ > l 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 7
Natioal Jio College Mathematics Depatmet 00 0 The mbes x satisfy the elatio x+ 7 x As, x s fo all positive iteges (i) Fid the exact vale(s) of s (ii) Show that if < x < 4, the x + < x The diagam shows the gaph of y e x The two oots of the eqatio x e x 0 ae deoted by a ad b whee a < b y x O a b x (i) Fid the vales of a ad b, each coect to decimal places A seqece of eal mbes x, x, x, satisfies the ecece elatio, x x + e fo (ii) Pove algebaically that, if the seqece coveges, the it coveges to eithe a o b (iii) Use the GC to detemie the behavio of the seqece fo each of the cases x 0, x ad x (iv) By cosideig x x +, pove that x x + > if x x + a < x < b, < if x < a o x > b (v) State biefly how the eslts i pat (iv) elate to the behavios detemied i pat (iii) 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 8
Natioal Jio College Mathematics Depatmet 00 Challegig Qestios Expess ( )( x ) x x i patial factios By sig the above eslt, show that b Hece fid a ad b sch that + b a N 4 N N ( ) Assigmet Qestios Fid, i tems of, the followig expessios: (a) ( + ) (b) ( + ) (c) ( 5 ) 0 A seqece of eal mbes satisfy the ecece elatio x + fo,,, x ( x + ) e (i) If x k as, fid the vale of k coect to decimal places (ii) Usig a gaphical method, show that if x > k, the x < + x j 0 j (i) Show that (ii) (iii) Hece fid k f ( ) 6 + 6 + + + + + ( )( + )( + ), whee k is a costat ( )( )( ), givig yo aswe i the fom State the sm to ifiity of the seies i which the th tem is + + + ( )( )( ) 4 Show that si( + ) x si( ) x cos( x) si x Hece, by sig the method of diffeece, show that, fo 0 < x < π, ( + ) N N x x cos( x) si cos si x Dedce that N x cos( x) cosec 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 9
Natioal Jio College Mathematics Depatmet 00 Nmeical Aswes to Seqeces ad Seies Ttoial Basic Mastey Qestios (a) 40 (b) 76 (c) ) 5 5, ; 5) 00 ( )( 5) 6 + + + Ttoial Qestios m (a) (m + )( m + ) (b) ( )(5 + 8) 6 (c) ( + ) log a + (a) 70 (b) 90 (i) ( + )(4 + ) (ii) ( )( + )( + ) ( + )( ), 4) +, (i) 5 56 (ii) 5) +, 6( ) 8( + ) 6( + ) ( + ), ( + )( + ) 8 6) 4950 7) ( +), + 8) 9 9(i) 79, + 0(i), 4 (i) a 069, b 5 Challegig Qestios a ad b 00 / SH / H Maths / Seqeces ad Seies (Teache s Editio) 0