UNIT 13: TRIGONOMETRIC SERIES

Transcript

1 UNIT : TRIGONOMETRIC SERIES UNIT STUCTURE. Larg Objctvs. Itroducto. Grgory s Srs.. Gral Thorm o Grgory s Srs. Summato of Trgoomtrc Srs.. CS Mthod.. Srs Basd o Gomtrc or Arthmtco-Gomtrc Srs.. Sum of a Srs of Ss (or Coss) of Agls Arthmtcal Progrsso.. Summato of Srs usg Bomal Srs.. Summato of Srs usg of Epotal Srs..6 Summato of Srs usg Logarthmc Srs ad Grgory s Srs.. Dffrc Mthod. Lt Us Sum Up.6 Aswrs to Chck Your Progrss. Furthr Radg.6 Modl Qustos. LEARNING OBJECTIVES Aftr gog through ths ut, you wll b abl to: kow about Grgory s srs dscrb th summato of trgoomtrc srs.. INTRODUCTION I prvous ut, w dscussd DMovr s Thorm ad ts som mportat dductos. W wll troduc Grgory s srs. Fally, w wll dscuss summato of trgoomtrc srs. Classcal Algbra ad Trgoomtry (Block ) 9

2 Ut Trgoomtrc Srs. GREGORY S SERIES Statmt: If ls wth th closd trval π π.., f, th ta ta ta ta π π,, s Proof: W hav ( ta) ( ) ( s) sc. Now, takg logarthm of both sds, w hav log( ta) log(sc. ) [log(ab) logalogb] logsc log log sc () Now, sc ls btw.., ta s umrcally ot gratr tha. W hav from () π π ad, ta ls btw ad, logsc log( ta ) (By Logarthmc srs (..) ta - ( ta ta ta ) ta (ta ) ta (ta ) to to ta ta ta (ta ) ( ) () Equatg magary part o both sds, w gt ta ta ta () s kow as Grgory s srs. Som Importat Dducto: ) Now w put ta So that ta Th w hav from () () ta whr 0 Classcal Algbra ad Trgoomtry (Block )

3 Trgoomtrc Srs Ut ) W quat th ral parts o both sds of (), w gt logsc ta ta ta Gral Thorm o Grgory s Srs Statmt: If ls btw π π ad π π.., π π π π, th, Proof: W put π ta ta ta π, th π Th gv codto rducs to Hc, ta ta(π ) s ta s sc. π π. Now, takg logarthm log( ta) log(sc. ) π π Θ π π ta(π π ) ta ta π ta ta(π ) log( ta ) ca b padd powrs of ta. log( ta ) ta ta ta ta (ta ta ) (ta ) to ta ta ta (ta ) ( )... log(sc ) Classcal Algbra ad Trgoomtry (Block ) (ta ) Equatg th magary parts o both sds, w hav

4 Ut Trgoomtrc Srs Or, ta ta π ta ta Eampl : Prov that: Soluto: L.H.S ta ta. π ( ) ( ) ( ) ta 6. π 6 π R.H.S Eampl : Sum th srs )... ).. Soluto: ) Th gv srs ) ( ) ( ) ( ) ta (By Grgory s srs ) Th gv srs..... Classcal Algbra ad Trgoomtry (Block )

5 Trgoomtrc Srs Ut ( ) ( ) ( ) (Grgory s srs)... ta Eampl : Prov that: π Soluto: R.H.S, ( π ). π ta ta (Grgory s srs) ta ta ta 9 ta ta. ta π R.H.S Eampl : Prov that π 8 ( ).9 Soluto: Th th trm of th srs s gv by ( ) T.9 ( ). Classcal Algbra ad Trgoomtry (Block )

6 Ut Trgoomtrc Srs T, T, T T Hc,, ad so o. S T T T T..... ta ta ta ta. ta ta ta ta. π L.H.S CHECK YOUR PROGRESS Q.: Prov that: Q.: Prov that: π Q.: Q.: π Prov that:.. ta If ls btw 0 ad π, prov that ta ta ta 6 ta 0 Classcal Algbra ad Trgoomtry (Block )

7 Trgoomtrc Srs Ut. SUMMATION OF TRIGONOMETRICAL SERIES Hr, w shall dscuss mportat mthods for summg up trgoomtrc srs whch may b ft or ft. Thr ar two mportat mthods for summato. Ths ar (a) mthod... C S Mthod Cosdr th srs: ad 0,a, a C C S mthod, (b) th dffrc a0 a ( β) a ( β) () S a0 s a s( β) a s( β) () Th abov srs may b ft or ft.th coffcts a... ad, β,... may b ay umbrs ral or compl. I th srs (), w hav trms whch cota s of umbrs.it s calld srs ad ts sum s dotd by C. Th srs () cotas ss of umbrs.it s calld s srs ad ts sum s dotd by S. C a Now, usg Eulr,s Thorm S a0( s) a[ ( β) s( β) ] [ ( β) s( β) ] ( β) ( β) a a a () C a 0 S a0( s) a[ ( β) s( β) ] [ ( β) s( β) ] ( β) ( β) a a a () 0 From th srs () ad (),w us C [(C S) (C S) ] ad S [(C S) (C S) ] to fd th valus of C ad S rspctvly. Classcal Algbra ad Trgoomtry (Block )

8 Ut Trgoomtrc Srs.. Srs Basd o Gomtrc or Arthmtco- Gomtrc Srs Sum of trms G.P a ar ar r a r or ar... ar r a r Sum of th ft Gomtrc srs: a ar ar ar... ar accordg as r < or r >. a, f r <.., < r <. r ar.. Sum of a Srs of Ss(or Coss) of Agls Lt Arthmtcal Progrsso { ( β} S s s( β) s( β) s( β) s ) W assum that, { ( β} C ( β) ( β) ( β) ) So, C S ( s) { ( β) s( β) } { ( β) s( β) } [ { ( ) β} s{ ( β} ] ) ( β) ( β) { β β ()β } β β β β. β β ( β ) ( β) β ( β β ( ) ) ( β) β { () β} { () β} { () β} ( s) { ( β) s( β) } { ( β) s( β) } [ { ( ) β} s{ ( ) β} ] ( β) 6 Classcal Algbra ad Trgoomtry (Block )

9 Trgoomtrc Srs Ut [ ( β) ( β) { () β} ] [ s s( β) s( β) s{ () β} ] ( β Equatg ral ad magary parts, w gt ( β) ( β) C ( β) ( β ) { ( ) β} [ { ( ) β} ] { ( β) ( β) } ( β) β β β β β s β β β β s β β β.s s β s β βs β s s s( β) s( β) s{ ( ) β} ad S ( β) [ s s{ ( ) β} ] { s( β) s( β) } ( β) s β β s β β β s s β β β β s Classcal Algbra ad Trgoomtry (Block )

10 Ut Trgoomtrc Srs β β s β.s s β s β s βs β s Hc, s s( β) s ( β) { ( ) β} { ( ) β} β s β s β s β β s β s () () Partcular cas (): Puttg β () ad (), w gt s s s s s s s ad s π β ) If β, th s sπ 0 () ad (), π π th, s s s to trms 0 ad π π to trms 0 Eampl : Sum to trms of th srs s s( β) s( β) s( β) Soluto: Lt S s s( β) s( β) s( β) to trms s s( π β) s(π β) s(π β) to trms 8 Classcal Algbra ad Trgoomtry (Block )

11 Trgoomtrc Srs Ut - s π β s s ( π β) s ( π β) ( -)( π β) ( π β) s β Eampl : Sum to trms of th srs: s s ( β) s ( β) Soluto: S s s ( β) s ( β) to trms - - ( β) - ( β) to trms ( to trms).β.s.β. s β. [ ( β) ( β) to trms] { (. ) β}.s( β) sβ Eampl : Sum th srs: to trms Soluto: C... to trms. Now, s s s S to trms ( s) ( s) ( s) C S sc ( sc ) sc sc ( sc )( sc ) ( sc )( sc ) sc to trms to trms Classcal Algbra ad Trgoomtry (Block ) 9

12 Ut Trgoomtrc Srs sc sc sc () sc sc sc sc ( ) sc sc sc ( sc ) () () sc sc sc sc sc sc ta sc sc [ ( ) s( ) ] sc ( s) sc( s) ta ( ) sc sc sc ta Equatg ral parts o both sds,w gt sc C sc sc sc sc s( ) sc s sc ta ( ) sc sc ta ( ) sc ta ( ) sc ta sc s ta [ ( ) ] ta ss ta CHECK YOUR PROGRESS Q.: Sum th srs: a) s s s to trms b) ( β) ( β) to trms c) to trms d) s s s to trms ) to trms 0 Classcal Algbra ad Trgoomtry (Block )

13 Trgoomtrc Srs Ut.. Summato of Srs usg Bomal Srs W should rmmbr th followg formula : ) Wh s a postv tgr ad,a ar ay compl umbr, ) w hav ( )! ( a) a ( )! ( a) a ( )! a a a ( )( )! ( ) a ( ) ( )! ( )( )! ( ) Wh s ay ratoal d ad s a compl umbr such that <, w hav ( ) ( ) ( ) ( )! ( )! ( )! ( )( )! ( )( )! ( )( )! ( ) Also, ( ) ( ) ad ( ) ( ) Eampl : Sum th Srs: Soluto: Lt Now, S s s s s.... s to. C to. Th, usg Boomal Thorm s to Classcal Algbra ad Trgoomtry (Block )

14 Ut Trgoomtrc Srs C S.. ( s) ( s) ( s) to ε [ ].... to to ( s)[ s s] ( s) ( s ) (s ) π π ( s ) ( s) s π π ( s ) ( s) s π π ( s ) s (Sc ( s)( φ sφ) ( φ) s( φ) ) π π ( s ) s Equatg th magary parts o both sds,w gt π ( s ) s S Eampl : Sum th srs Soluto: Lt Th, C S s s s C S...6 ( s) ( s) ( s ) Classcal Algbra ad Trgoomtry (Block )

15 Trgoomtrc Srs Ut... whr...6 ( ) ( ) ( s) s s s [By D Movr s Thorm] Equatg ral parts, w gt C.. Summato of Srs usg of Epotal Srs Th followg srs ar frqutly usd. )!! ) ) v)!! s!!!! Classcal Algbra ad Trgoomtry (Block )

16 Ut Trgoomtrc Srs v) sh!! v) h!! Eampl : Sum th Srs Soluto: Lt ad Th, C S β C β S sβ β! β! sβ! (β sβ)! β! β! sβ! (β sβ)! ( β sβ) β β β β ( β s β).! s β β! β { ( sβ) s( c sβ) } Equatg ral parts o both sds, w gt C β ( sβ) Eampl : Fd th sum of th srs Soluto: Lt ad c c!! c c C!! c c S s s!! c! Th C S ( s) ( s) c! c! ( ) h c { ( ) } h c s c! Classcal Algbra ad Trgoomtry (Block )

17 Trgoomtrc Srs Ut { c( s) } ( c c s) h( c ) ( c s) sh( c ) s( c s) Equatg ral parts, w gt C h( c ) ( c s)..6 Summato of srs usg Logarthmc srs ad Grgory s srs ) log( ) ) log( ) ) ta whr Eampl: Sum th Srs: ) ) s s s Soluto: Lt th srs () ad () b dotd by C ad S rspctvly.. C ad S s s s Th C S ( s) ( s) ( s ) log( ) by Logarthmc srs log( s) log ( { ) s } ta s s log( s ) ta Classcal Algbra ad Trgoomtry (Block )

18 Ut Trgoomtrc Srs log log ( ) ta s { ( ) } ta ta log. ta log. log. ta ta ta Equatg ral ad magary parts, w gt C log.. ) log ad. ) s s s S CHECK YOUR PROGRESS Q.6: Sum th srs:! ) ( β) ( β) ) ) v) β β!! s s s.. Dffrc Mthod 6 I ordr to sum a srs,somtms t s covt to splt up ach trm as th dffrc of two prssos such that o prsso of ach dffrc occurs succdg wth a oppost Classcal Algbra ad Trgoomtry (Block )

19 Trgoomtrc Srs Ut sg. Th splttg s do such a way that wh all th trms of th srs ar addd togthr, th compot trms cacl pars. Fally w ar lft wth two trms, o from th frst trm ad o from last trm. Suppos, w hav to fd th sum of u u u u Frst w wrt f( ) f() () u From () puttg u u u,,,...,, w gt f() f() f() f() f() f() u f( ) f() Addg vrtcally, w gt u u u u f( ) f() (sc all th trmdat trms cacl pars) Eampl: Sum th srs to trms ta Soluto: Hr T ta ta ta ta ta ta ( ) ( ) ( ) ta () Now, T ta ta T T ta ta ta ta T ta Addg, w gt ( ) ta Eampl : Sum th srs: Θ ta () ta ta y ta ( ) y y S T T T T ta ( ) ta ssc ssc s sc to trms Classcal Algbra ad Trgoomtry (Block )

20 Ut Trgoomtrc Srs Soluto: W hav s s ssc. T s s( ) s s ta [ ta ] Smlarly, T [ ta ta] [ ta ta ] T [ ta ta ] T [ ta ta ] T Addg ths,w hav th rqurd sum S T T T T [ ta ] ta, othr trms cacllg ach othr. CHECK YOUR PROGRESS Q.: Sum th srs to trms by Dffrc Mthod. a) ta ta ta ta 9 6 ( ) b) ss s s s s c) tata ta ta ta ta 8 Classcal Algbra ad Trgoomtry (Block )

21 Trgoomtrc Srs Ut. LET US SUM UP If ls wth th closd trval th srs ta ta ta π π,,.., f ta π π, th s calld Grgory s srs.w also drv mportat rsults from Grgory s srs. W dscuss mportat mthods for summg up trgoomtrc srs whch may b ft or ft.thr ar two mportat mthods for summato.ths ar (a) C S mthod (b) Th dffrc mthod..6 ANSWERS TO CHECK YOUR PROGRESS As. to Q. No. : W hav ta.. π π 8 As. to Q. No. : From R.H.S, Classcal Algbra ad Trgoomtry (Block ) 9

22 Ut Trgoomtrc Srs ta ta (By Grgory s srs ) ta ta ta ( ) ta 9 ta ta. π Th L.H.S As. to Q. No. : L.H.S - ad.f.. ad.f.. ta By Grgory, s srs bcaus <. R.H.S As to Q No : W hav - ta ta s ta ta () π Sc ls btw 0 ad π ls btw ad,so that ta <. - ta ta π Sc ls btw 0 ad s ta π ls btw ad,so that ta ta <. 60 Classcal Algbra ad Trgoomtry (Block )

23 Trgoomtrc Srs Ut Thrfor, ta ta ca b padd by Grgory s srs. ta ta ta ta 6 0 ta - ta ta - Hc, from () ad (), w hav ta () - ta ta As. to Q. No. : a) Lt S b) Lt S - ta 6 ta 0 - Hc provd. s s s s 6 () s s ( ) sc ( 6 ) ( ( β) ( β) ( β) ) ( ) {( ( β) } { ( β) } { ( ( } ) [ ( β) ( β) ( ( ) ] ( ) β (β) s β s ( ( ) β)sβ cβ c) Lt S ( ) ( ) ( 6) ( ) Classcal Algbra ad Trgoomtry (Block ) 6

24 Ut Trgoomtrc Srs [ 6 ] () s s d) Try yourslf Lt S ( ) s s s ) Try yourslf Lt S s s ( ) s s s ( s s s ) s s As. to Q. No. 6: ) Lt C ( β) ( β)!! ad S s s( β) s( β) th C S! ( s) [ ( β) s( β) ] [ ( β) s( β) ]! ( β) ( β) β. ( ). β β ( β s β) β.! ( s β) β [ ( sβ) s( sβ) ] Equatg ral parts o both sds, β w gt C.( sβ) ths s a potal srs 6 Classcal Algbra ad Trgoomtry (Block )

25 Trgoomtrc Srs Ut C S C S ) Lt C β β ) S sβ sβ s ( β sβ) ( β sβ) ( β sβ) Lt β β β ( s) ( β) s S s ( β) ( β) s( β)! C!!! ( β) ( β)! [ ( β) s( β) ] [ ( β) s( β) ] ( β)! β!!! ( β) β β ( ) By srs ( s) (β sβ) ( s)[ (β)(sβ) s(β)s(sβ) ] ( s)[ (β)h(sβ) s(β)sh(sβ) ] ( s)[ (β)h(sβ) s(β)sh(sβ) ] ( β)h(sβ) s( β)sh(sβ) ( β)h(sβ) s( β)sh(sβ) s ( β)h(sβ) s s(β)sh(sβ) [ (β)h(sβ) s s(β)sh(sβ) ] Equatg magary parts, w gt S s ( β)h(s β) s( β)sh(s β) ) Lt C!! ad S s s s!! Classcal Algbra ad Trgoomtry (Block ) 6!

26 Ut Trgoomtrc Srs! Th ( ) ( ) ( s) C S ( ) s! ( s ). s! s! By potal srs [ ( s) s( s) ] ( ) s s ( s ) Equatg th ral parts o both sds,w gt C ( s) v) Lt S s s s ad C Th C S ( s) ( s) ( s) log ( ) ( ) log s log ( ) ( s) By Logarthmc srs ta Equatg magary parts, w gt S ta As. to Q. No. : a) Hr Puttg s (Sparatg ral ad magary parts) s (cpt wh ),,,..., T ta T ta T ta ta ( ) ta ta ta ta ta ( ) ( ) ta, w hav ( ) 6 Classcal Algbra ad Trgoomtry (Block )

27 Trgoomtrc Srs Ut T ta ta T ta ( ) ta Addg, w gt th rqurd sum to trms S T T T T ta ( ) ta ( ) ta ta b) Hr, ss.ss T T [ ] s s.s s [ ] T T Addg up th abov rlatos, w th rqurd sum S T T T T d) Try yourslf. R qurd sum ta ta. FURTHER READING ) Durll, C. V. ad Robso, A; Advacd Trgoomtry. ) J. Smth, Karl; Esstals of Trgoomtry. ) N. Aufma, Rchard, C. Barkar, Vro & D. Nato, Rchard; Collg Algbra. Classcal Algbra ad Trgoomtry (Block ) 6

28 Ut Trgoomtrc Srs.8 MODEL QUESTIONS Q.: Fd th sum of th followg srs:! a) s s( β) s( β) b) s. s s c) s s s d) ) 9 - f) ta ta ta to trms 9 g) - ta ta ta ta 9 9 *** ***** *** 66 Classcal Algbra ad Trgoomtry (Block )

29 REFERENCES ) Bal, N. P.; Trgoomtry, for B.A/B.Sc. Classs (st Edto), Nw Dlh: Lam Publcato (P) Ltd. ) Ksha, Har (00); Trgoomtry; (st Edto), Nw Dlh: Atlatc Publshrs. ) Lpschtz, Symour; Lar Algbra: Schum Solvd Problms Srs; Tata McGraw Hll. ) Mapa, S. K.; Hghr Algbra (Classcal). ) Ray, M. & Sharma, H. S.; A Tt Book of Hghr Algbra. 6) Vasstha, A.R. & Sharma, S. K.; Trgoomtry. ) Vasstha, A. R.; Matrcs; Mrut: Krsha Prakasha Madr. Classcal Algbra ad Trgoomtry (Block ) 6