Chpter Ifiite eries Pge of 9 Chpter : Ifiite eries ectio A Itroductio to Ifiite eries By the ed of this sectio you will be ble to uderstd wht is met by covergece d divergece of ifiite series recogise geometric series determie whether give geometric series coverges d determie its sum A Itroductio A lot of studets e cofused betwee sequeces d series. A sequece is of the form,,, 4,... A series is of the form + + + 4 +... For exmple,,, 4,... is exmple of sequece but + + + 4 +... is exmple of series. We use the Greek letter,, proouced sigm for writig the series i compct form. For exmple + + + 4 +... is writte s, tht is + + + 4 +... How would you write the followig i, sigm, ottio? ) + + + + +... ) + + 4 + 8 + 6 +... ) + + + +... ) c be writte s + + + + +... + + + + + ( ) ) c be writte s 4 8 6... [tt t becuse ] ) c be writte s + + + +... + These ), ) d ) e ll exmples of ifiite series. Wht does ifiite series me? A series which hs ulimited umber of terms is clled ifiite series. A sequece,,,..., defied by [um of the First Term] + + + [um of the First Terms] [um of the First Terms] + + + + [um of the First Term]...
Chpter Ifiite eries Pge of 9 e clled the th ptil sum of the series. These e exmples of fiite series. Wht does fiite series me? A series which hs fiite umber of terms is clled fiite series. I this chpter we cosider ifiite series d e iterested i the covergece of these ifiite series. Wht does coverget series me? Let be the th ptil sum of ifiite series which mes it is the sum of the first terms: + + + +... If pproches limit L s the umber of terms,, pproches ifiity the we sy the ifiite series coverges. Tht is ifiite series is coverget if lim L ( ) where L is rel umber. If coverges to rel umber L s goes to ifiity the the series is sid to coverge to L d L is lso clled the sum of the series. For exmple + + + +... 5 5 5 5 4 This mes ddig ifiitely my terms,,,, etc gives the sum 5 5 5 5 4. I the tble below we hve evluted the sum i the right colum for correspodig umber of terms i the left colum. k k 5. 5.499.499999744.5 TABLE We c sum fiite umber of the terms by usig computer lgebr system such s MAPLE. If lim does ot pproch rel umber s goes to ifiity the the series is ( ) sid to diverge. For exmple + + + +... diverges k k Wht does this me? If we tke eough terms of the series the we c mke the sum s lge s we like. k k.9896854 5.877758 7.4854786 9.787666.946 4.9767 TABLE
Chpter Ifiite eries Pge of 9 I the tble bove the left hd colum is the umber of terms () d the right colum gives the correspodig sum: Tht is the sum of the first terms of the series k k is just over 5 d the sum of the first millio terms ( ) is bout 4.4. Geerlly for the rest of this chpter we test the covergece of give ifiite series. For this test we eed to kow certi results regdig sequeces to show tht series coverges. For exmple we use the followig result to test covergece: lim x if x < (.) ( ) If x the lim ( x ) does ot coverge. I testig give series for covergece we first write th ptil sum, determie lim ( ) by usig our results of sequeces from the lst chpter., d the To get feel for series it is good prctice to write out the first few terms of the series. Exmple how tht olutio. k k coverges d fid its sum. We first cosider the th ptil sum, Well k k, of k k. How c we write? is the sum of the first terms d is writte s: + + + +... + * Multiplyig both sides of (*) by gives + + + +... + Multiplyig ech term + + + +... + i brcket by implifyig by + + + +... + Rules of Idices ubtrctig (**) (*) gives: + + + +... +... + + + + + + + +... +... ( ) Becuse subtrctig ALL middle terms gives (**)
Chpter Ifiite eries Pge 4 of 9 Hece you e oly left with the first d lst terms of. ice therefore the sum of the ifiite series, k is give by k lim ( ) lim ubstitutig How do we fid the vlue of lim? Becuse lim lim by lim x if x < (.) ( ) Therefore lim lim lim. We sy the give series coverges d its sum is d this is Tht is ( ) writte more formlly s: Wht does this, k k, me? k k Addig ifiitely my terms of the form (d ech of them positive) gives the k sum equl to or + + + +... A Geometric eries The bove is exmple of geometric series. The geerl geometric series is give by. Wht e the first few terms of the series? 4 5 + + + + + + ( ) + + + + +... +...... Wht do we me by geometric series? A ifiite series i which ech term is obtied from the precedig term by multiplyig by r. For exmple the secod term,, is obtied from the first term,, by multiplyig it by costt r. The symbol r is clled the commo rtio d is clled the first term. I Exmple wht is r d equl to? Commo rtio r d the first term. Next we test this series for covergece.
Chpter Ifiite eries Pge 5 of 9 Exmple how tht the geometric series,, coverges for r < d diverges for r. olutio. Let be the th ptil sum of the geometric series. How c we write is the sum of the first terms of the give series: + + + + +... + Assume r. Multiplyig both sides by the commo rtio r gives r + r + r + r + r +... + r ubtrctig? ( ) ( ) ( ) ( ) ( )... [ Usig the Rules of Idices] + + + + + + r gives r + + + + +... + r ( ) ( ) (... ) + + + + + + r r ( ) ( r ) [ ubtrctig All other terms give ] Tkig out Commo Fctors from both sides r r [ Dividig both sides by r ] Multiplyig Numertor r d Deomitor by If r < the the sum of the geometric series, becuse lim ( r ) (.) ( x ), is give by ( ) ( ) r r lim ( ) lim ubstitutig r r ( lim( r )) r ( ) r r by lim if x < Hece the geometric series coverges for r <. Does the series coverge for r? r the lim ( r ) If diverges d therefore the sum of the geometric series ( r ) lim ( ) lim diverges r
Chpter Ifiite eries Pge 6 of 9 The geometric series coverges for r < d the sum is but it diverges for r. r ummizig the bove exmple we hve tht the geometric series is the ifiite series defied s (.) It is coverget if r < d the sum is (.) + + + +... + +... ( ) r [ First Term] [ ] Commo Rtio If r the the geometric series diverges. A Exmples of Geometric eries Exmple how tht the followig ifiite series coverges d fid its sum. olutio Writig out we hve + + + +... Wht type of series is this? It is geometric series becuse ech term is r d the first term. Does the series coverge? Yes becuse the modulus of the commo rtio, (.) if r < r tht coverges d the sum is the previous term. The commo rtio r < so we c coclude by
Chpter Ifiite eries Pge 7 of 9 This mes ddig the ifiite umber of terms of the form to. Tht is + + + +... gives the sum equl Exmple 4 how tht the ifiite series coverges d fid its sum. olutio Is the give series geometric series? Writig out the series iformlly we hve x where x < x x x x x x + + + + + + x+ x + x + x +... Ech term is obtied by multiplyig the precedig term by x. Therefore yes it is geometric series. Wht is the first term,, d the commo rtio, r, equl to? First term d the commo rtio r x d becuse r x < therefore by (.) the series provided x <. if r < r x coverges. The sum is give by x x... First Term ( Commo Rtio) Note tht the lower limit stts t. First elemet of series my ot stt t but or or 69 d it is deoted by, or 69 Exmple 5 Determie whether the followig series is coverget: + +... 9 7 If it is coverget the fid its sum.
Chpter Ifiite eries Pge 8 of 9 olutio Is the give series geometric series? If we divide two cosecutive terms the we hve: [ ecod Term] [ First Term] or [ Third Term] [ ecod Term ] or 9 [ Fourth Term] [ Third Term] 7 9 Hece there is commo rtio, r, betwee two cosecutive terms. Normlly it is esier to divide the first two terms of the series. o we hve geometric series with commo rtio therefore by (.) r < if r < r r. ice the give series coverges d the sum is equl to First Term + +... 9 7 ( Commom Rtio ) 7 4 4 This mes sum of the ifiite series is 7, tht is + +... 7 9 7 Exmple 6 Determie whether the followig series is coverget: ( ) If it is coverget the fid its sum. olutio Writig out the first few terms of the series we hve ( ) ( ) Becuse + + + +... Is this geometric series?
Chpter Ifiite eries Pge 9 of 9 Yes becuse ech term is obtied by multiplyig the precedig term by the commo rtio r. Does the series coverge? No becuse r therefore the series diverges. Wht does this me? Addig ifiitely my terms + + + +... does ot pproch limit. UMMARY A ifiite series is deoted i compct form s s: Geometric series is defied s + + + 4 +... d c be iformlly writte where is the first term d r is the commo rtio. This series coverges if r < d the sum is give by (.) But the series diverges for r. [ First Term] [ ] r Commo Rtio