UTCLIFFE NOTE: CALCULU WOKOWKI CHAPTER Ifiite eries Coverget or Diverget eries Cosider the sequece If we form the ifiite sum 0, 00, 000, 0 00 000, we hve wht is clled ifiite series We wt to fid the sum of this ifiite series, if possible I order to do this, we cosider the sequece of prtil sums, where k is the sum of the first k terms of the ifiite series For the give exmple we hve, 0 0 00 0 0 00 000 0 d so o Thus, the sequece of prtil sums 0 00 000 0000 0 is 0, 0, 0, 0, which we c show lter will coverge to / Defiitio A ifiite series (or series) is expressio of the form, which c be writte usig the summtio ottio s or Ech umber k is term of the series, d is the th term Defiitios (i) The kth prtil sum k of the series k k (ii) The sequece of prtil sums of the series,,,, k is is
Defiitio A series ie, if is coverget (or coverges) if its sequece of prtil sums coverges, lim for some rel umber The limit is the sum of the series d we write The series is diverget (or diverges) if diverges A diverget series hs o sum ome Importt eries Exmple () Fid,,,,, 6 (b) Fid (c) how tht the series coverges d fid its sum olutio: () By defiitio, 6 6 0 6 6 6 6 6 6 7 6 6 7 7 (b) Usig prtil frctios, oe c verify tht o we my write the th prtil sum s Thus, we hve
(c) lim lim The series coverges d hs the sum This series is exmple of telescopig series Exercise Fid (),,, (b), (c) the sum of the series, if it coverges # 7 olutio: () 7 7 7 7 7 7 (b) A B 7 7 (d) A 7 B 7 : B B : A A 7 7 7 7 7 7 7 7 7 lim lim 7 7 7 Exmple () Fid,,,,, 6 (b) Fid (c) how tht the series diverges olutio:, 0,, 0,, 0 () 6 if is odd (b) 0 if is eve (c) lim does ot exist
Exmple Prove tht the series olutio: Group the terms of the series s follows: is diverget 6 7 8 9 6 7 Note tht ech group hs twice s my terms s the precedig group Also, sice icresig the vlue of the deomitor decreses the vlue of frctio, we hve Thus, we get 8 6 6 7 8 8 8 8 8 9 0 6 6 6 6 7 8 6 7 8 We c prove by mthemticl iductio tht k k for every positive iteger k This mes tht we c mke s lrge s we wt by tkig sufficietly lrge; ie lim ice the sequece of prtil sums diverges, the give series diverges Defiitio The hrmoic series is the diverget series Defiitio The geometric series is the series give by r r r r
where d r re rel umbers, with 0 Theorem The geometric series (i) coverges d hs the sum r if r (ii) diverges if r Proof: If r, the d lim d the series diverges If r, the if k is odd k 0 if k is eve d sice the sequece of prtil sums oscilltes betwee d 0, the series diverges If r, the r r r d r r r r r ubtrctig correspodig sides of the equtio we obti olvig for r r, we get r r r r r Hece, r r lim lim lim lim r r r r r r r If r, the lim r 0 d hece lim r If r, the lim r does ot exist d hece lim does ot exist d the series diverges Exmple how tht the series 0 00 000 0 coverges d fid its sum olutio: The give series is geometric with 0 d r or 0 ice r, the series coverges d the 0 0 0 sum is r 0 09 9 This is oe wy to show tht 0
#8 olutio: This is geometric series with d r ice r r, the series coverges d the sum is e e #0 olutio: This is other geometric series with r e e e e d r ice r, the series coverges d the sum is # olutio: This is geometric series with d r ice r, the series diverges I the followig, fid ll the vlues of x for which the series coverges, d fid the sum of the series #8 x x x olutio: Note tht the series is geometric series with d r x We kow tht for this series to coverge we must hve r x d thus we must hve x d the series will coverge to x #0 x x x olutio: This is gi geometric series with d r x For the series to coverge, we must hve x olvig this, we get x x The sum is give by 9 9 r x x x 6
I the followig, express the repetig deciml s series d fid the rtiol umber it represets # 6 olutio: 6 6 6 666 06 00006 000 000000 Igorig the first term, we hve geometric series with 6 000 6 6 999 999 000 6 d r o we hve 000 000 Theorem If series Proof The th term, the lim is coverget, the lim 0 c be expressed s ice we kow tht coverges to sum, sy d lso lim Thus, lim lim lim lim 0 Wrig! The theorem sys tht if we kow tht the series coverges, we my coclude tht the th term goes to 0 s becomes lrge The coverse is ot ecessrily true! A good exmple is the hrmoic series which is diverget lthough its th term 0 th-term Test (i) If lim 0, the the series is diverget (ii) If lim 0, further ivestigtio is ecessry to determie covergece or divergece of # 0 olutio: lim 0 D 0 0 #6 e olutio: lim 0 o coclusio e 7
#8 si olutio: lim x si 0 x x si ' cos 0 LH lim x lim x x lim cos 0 D x 0 x x x x x Theorem If d b re series such tht j bj for every j > k, where k is positive iteger, the both series coverge or both diverge Proof: Let k k b b b bk k Let d T deote the prtil sums of d b, respectively It follows tht if k, the, k T Tk or T T Thus, k k lim lim T T, k k d so either both limits exist or both do ot exist If both series coverge, the their sums differ by T k k Note: Chgig fiite umber of terms of series does ot ffect the covergece or divergece of the series (lthough it does chge the sum!) o, we c chge the first k terms to 0 d the covergece or divergece of the resultig series is ot ffected Chgig the first k terms to 0 is the sme s deletig the first k terms of the series Theorem For y positive iteger k, the series d k k k either both coverge or both diverge Determie whether the series coverges or diverges #6 0 9 0 olutio: This is the telescopig series (Exmple ) with the first 9 terms deleted ice the origil series coverges, the this ew oe lso coverges 0 8
Theorem If d b re coverget series with sums A d B, respectively, the (i) b coverges d hs the sum A+B (ii) c coverges d hs the sum ca for every rel umber c (iii) b coverges d hs the sum A-B Note: If diverges, the c lso diverges for c 0 Theorem If Proof: is coverget d b is diverget, the b Write b b d do idirect proof is diverget Determie covergece or divergece If coverget, fid its sum # olutio: D D C # olutio: both coverget G C 6 9