MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2 is the secod term, d i geerl is the th term. We usully write isted of the fuctio ottio f ( ) for the vlue of the fuctio t the umber. The sequece {, 2, 3,... } is lso deoted by { } or { } =. Exmple: Some sequeces c be defied by givig formul for the th term. I the followig exmples we give three descriptios of the sequece. Notice tht does t hve to strt t. By usig proceedig ottio By usig the defiig formul By writig out the terms of the sequece () + = = 2 3,,,...,,... + 2 3 4 + (b) ( ) ( + ) (c) 3 { 3} = 3 3, 3 ( ) ( + ) 2 3 4 5 ( ) ( + ) = 3,,,,...,,... 3 9 27 8 3 = { 0,, 2, 3,..., 3,... }
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 2 Exmple: Here re some sequeces tht do t hve simple defiig equtio. () The sequece { p }, where p is the popultio of the world s of Jury i the yer. (b) The Fibocci sequece { f } is defied recursively by the coditios f = f = f = f + f 3 2 2 Ech term is the sum of the two precedig terms. The first few terms re {,, 2,3, 5,8,3, 2,... } This sequece rose whe the 3 th -cetury Itli mthemtici kow s Fibocci solved problem cocerig the breedig of rbbits. Defiitio : A sequece { } hs the limit L d we write lim = L or L s if we c mke the terms { } s close to L s we like by tkig sufficietly lrge. If lim exists, we sy the sequece coverges (or is coverget). Otherwise, we sy the sequece diverges (or is diverget). Theorem : If lim ( ) = d f ( ) If lim diverges to. f x L x =, the the sequece { } = whe is iteger, the lim = L. is diverget but i specil wy. We sy tht { } The Limit Lws give i Chpter 2 lso hold for the limits of sequeces. Aother useful fct bout limits of sequeces is give by the followig theorem. Theorem 2: If lim = 0, the lim = 0
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 3 Exmple : Fid lim +. Exmple 2: Evlute lim ( ) if it exists.
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 4 Sectio 7.2 Ifiite Series If we try to dd the terms of ifiite sequece { } = + 2+ 3+ + + we get the expressio of the form which is clled ifiite series (or just series) d is deoted, for short, by the symbol or = To determie whether or ot geerl series, + 2+ 3+ + +, hs the sum, we cosider the prtil sum. s 2 2 3 2 3 = s = + s = + + s = + + + + 2 3 These prtil sums form ew sequece { } s which my or my ot hve limit. If lim s = s exists (s fiite umber), the we cll it the sum of the ifiite series =. Defiitio : Give series = + 2 + 3+, let s deote its th prtil sum: If the sequece { s } series s = = + + + + i 2 3 i= is coverget d lim s = sexists s rel umber, the the is clled coverget d we write + + + + + = s or = s 2 3 The umber s is clled the sum of the series. Otherwise, the series is clled diverget. =
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 5 Thus the sum of series is the limit of the sequece of prtil sums. So whe we write = = s, we me tht by ddig sufficietly my terms of the series we c get s close s we like to the umber s. Notice tht = lim = i= i Exmple: Determie whether the series + + + + + + 2 4 8 6 2 is coverget or diverget. If it is coverget, fid its sum. Solutio: Cosider the prtil sum: s s s s = 2 3 = + = = 2 2 4 2 7 = + + = = 2 2 2 8 2 2 2 2 3 2 3 3 = + + + + = 2 3 2 2 2 2 2 As we dd more d more terms, these prtil sums become closer d closer to. I fct, by ddig sufficietly my terms of the series we c mke the prtil sums s close s we like to. So it seems resoble to sy tht the sum of this ifiite series is d to write lim s = lim 2 = = + + + + + + = 2 2 4 8 6 2 = Hece the series is coverget d its sum is. Note: We c lso sy tht Tht is, the sequece { s } coverges to. lim s = lim = 2
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 6 The geometric series 2 3 r r r r r = = + + + + + + 0 is coverget if r < d its sum is = r = r < r If r, the geometric series is diverget. Exmple 2: Fid the sum of the geometric series 0 20 40 2 5 + + = 5 3 9 27 = 3 2 Exmple 3: Is the series 2 3 coverget or diverget? =
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 7 Exmple 4: Show tht the series is coverget, d fid its sum. ( ) = + Defiitio : The hrmoic series is diverget series of the form = = + + + +... 2 3 4 Theorem : If the series = is coverget, the lim = 0
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 8 Sectio 7.3 Covergece Tests I this sectio we will develop vrious tests tht c be used to determie whether give series coverges or diverges. 7.3. The Divergece Test Theorem: (The Test for Divergece) () If lim does ot exist or if lim 0, the the series = is diverget. (b) If lim = 0, the the series Exmple : Show tht the series = 2 = 5 + 4 my either coverge or diverge. 2 diverges. Theorem 2: If d b re coverget series, the so re the series (where c is costt), ( + b), d ( b), d c (i) c = c = = (ii) ( ) + b = + b = = = (iii) ( ) b = b = = =
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 9 Exmple 2: Fid the sum of the series 3 + ( ) 2 = +. Note: A fiite umber of terms does t ffect the covergece or divergece of series. For istce, suppose tht we were ble to show tht the series is coverget. Sice 3 = 4 + 2 3 = + + + + 2 9 28 + 3 3 = = 4 it follows tht the etire series is coverget. Similrly, if it is kow tht the 3 = + series = N+ coverges, the the full series N = + = = = N+ is lso coverget.
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 0 7.3.2 The Itegrl Test Theorem 3: (The Itegrl Test) Suppose f is cotiuous, positive, decresig fuctio o [, ) series other words: d let f ( ) is coverget if d oly if the improper itegrl f ( ) = =. The the x dxis coverget. I (i) If f ( ) x dx is coverget, the = is coverget. (ii) If f ( ) x dx is diverget, the = is diverget. Note: Whe we use the Itegrl Test, it is ot ecessry to strt the series or the itegrl t =. For istce, i testig the series we use ( 3) 2 = 4 4 ( ) 2 3 dx Also, it is ot ecessry tht f be lwys decresig. Wht is importt is tht f be ultimtely decresig, tht is, decresig for x lrger th some umber N. The = N is coverget, so = is coverget. Exmple 3: Test the series for covergece or divergece. 2 = +
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Theorem 4: The p-series = p is coverget if p > d diverget if p. Exmple 4: Determie whether the followig series is coverget or diverget. = + + +... (i) 3 3 3 3 = 2 3 = = + + +... 2 3 (ii) /3 3 3 3 3 = =
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 2 7.3.3 The Compriso Test Theorem 5: (The Compriso Test) Suppose tht d (i) If b is coverget d (ii) If b is diverget d b re series with positive terms. b for ll, the b for ll, the is lso coverget. is lso diverget. I the compriso tests the ide is to compre give series with series tht is kow to be coverget or diverget. I usig Compriso Test we must, of course, hve some kow series b for the purpose of compriso. Most of the time we use oe of these series: (i) A p-series ( = p is coverget if p > d diverget if p ) (ii) A geometric series ( r coverges if = r < d diverges if r ) 5 Exmple 5: Determie whether the series coverges or diverges. 2 = 2 + 4+ 3
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 3 7.3.4 The Limit Compriso Test Theorem 6: (The Limit Compriso Test) Suppose tht d b re series with positive terms. If lim b = c where c is fiite umber d c > 0, the either both series coverge or both diverge. Exmple 6: Test the series for covergece or divergece. 2 = 2 2 + 3 Exmple 7: Determie whether the series coverges or diverges. 5 = 5 +
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 4 7.3.5 The Rtio Test Theorem 7: (The Rtio Test) Let be series with positive terms d suppose tht lim + ρ = (i) If ρ <, the the series = (ii) If ρ > or ρ =, the the series is coverget. = is diverget. (iii) If ρ =, the Rtio Test is icoclusive; tht is, o coclusio c be drw bout the covergece or divergece of. = Exmple 8: Test the series 3 3 = for covergece. Exmple 9: Test the covergece of the series. =!
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 5 7.3.6 The Root Test Theorem 8: (The Root Test) Let be series with positive terms d suppose tht (i) If ρ <, the the series ρ = lim = lim( ) / = is coverget. (ii) If ρ > or ρ =, the the series = is diverget. (iii) If ρ =, the Root Test is icoclusive. Exmple 0: Test the covergece of the series = 2 + 3 3 + 2
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 6 Sectio 7.4 Tylor d Mcluri Series Defiitio : If f hs derivtive for ll order t, the we cll the series = 0 ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) f f f f f ( x) = x = f + x + x + x +...!! 2! 3! the Tylor series of the fuctio f t (or bout or cetered t ). For the specil cse = 0 the Tylor series becomes = 0 ( ) ( 0) ( 0) ( 0) ( ) ( ) 2 f f f f ( x) = x = f 0 + x+ x +...!! 2! This cse rises frequetly eough tht it is give the specil me Mcluri series. f x = e. Exmple : Fid the Mcluri series of the fuctio ( ) x
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi 7 Note: e x x = = 0! for ll x e = = + + + +...!! 2! 3! = 0 f x Exmple 2: Fid the Tylor series for ( ) x = e t = 2. Exmple 3: Fid the Mcluri series forsi x.