Lecture Note
University
California State University, Los AngelesCourse
MathematicsPages
11
Academic year
2023
yung dump
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I 10.6 Alternating Series The alternating series has the form 8 00 San = Ecy n+1 Un = U1- U2 + U3 - Uy n=1 n=1 Question: when conv ? Th (Alternating Series Test - -AST) conv. if 1 Un >0 Hn and 2 Unti v Un for large n V Un 1) and 3 lim Un = 0 n 8
2 Exp ( Alternating Harmonic Series) 00 n+l E = + n 2 3 4 n=1 Un = lan \ 1) Un = lanl >0 2 Un = \ n 3 lim Un = lim \ =0 n 00 n 00 n 8 Hence, W n+1 (-1) n Conv. by AST n=1 Exp ( Alternating Geometric Series) 8 2 3 2 2 E n = - 2 (-1) (0.2) + + 10 10 n=1 2 We Know it Conv. to 10 1- 2 10 6) 1) Un = lanl >0 un 2 Un = 10 )" 10.1 Th5 3 lim Un = lim (0.2) =0 n 8 n 8 00 Hence, E (-1) (0.2)" Conv. by AST n=1
3 0 Exp { (ii)n this alter nating series div n=1 by the nth term test Since lim Un = lim n #0 n 8 n 00 Exp W (-1) n : :, = = n+ 3 Is = = = 2 lim Un = lim n I # 0 n 8 n 00 n+3 pet ( Conv. Abs.) The infinite Series San conv. Abs. if [ anl conv. 8 n+l Exp (-1) n2 conv. by AST since II) Un>0 2 Un n = l 3) lim 1/20 =0 n 00 Is it converging Absolutely ? 8 8 ntl Elant (-1) nz so n2 n=1 n=1 conv. Abs. Conv. p- serie
3.1 Th If Elanl Conv. then Ean Conv. VL This means if a series conv. Abs. then it converges 8 n+1 Exp [-1) 1 "Alternating Geometric Series" n n=o 2 8 {lanl = { I = I + 1/2 + = =2 2 2 22 I n=o n+1 El an) converges to 2 Ean = + conv. 2n "byAST" Remark The Converse of Th above is not True : Means: if Ean conv. [lan) conu. nt' Exp { (-1) "Alternating Harmonic Series" n n=1 This series converges by AST but not Abs. since D { an = E which is the divergent Harmonic n=1 Series. This Remark says: If a series converges, then it may not converges Abs.
4 0 n+1 Exp Does W (-1) I Conv. Abs. ? n n=1 This is the alternating harmonic Series we have seen that it Conv. by AST But not Abs. since 00 Elant which is the n divergent harmonic n=1 series. pet ( Converge Conditionally) The in finite series Ean Conv. Cond, if it Conv. by AST but not Abs. see exp. Above n+l Exp (ti) I conv. cond. n n=1 n Exp EN (-1) Vn conv. by AST since DV (2)2 3 n=1 but not Abs since Elanl Un This series Conv. Condit. div. p- series
5 8 n+) Remark (-1) I = conv. Abs. if p> / n° conv. condi if OCPEI n=1 Th (Alternating Estimation Th) 00 8 Assume Ean = [t" n+l Un = U1- U2 + U3 + L n=\ n=1 n+l If we approximate L by Sn = U,- U2 + U3 (1)Un then I the remainder L_ Sn has same sign as a n+1 2 the error IL - Sn I < unt n+1 janti I Vn 3 min { Sn, Su+1} L < max {Sn, Sn+1}
6 8 n+l n 3 4 Exp [t" 3) 3 = 2 3 3 3 2 3 n=1 3/3 = )- 7/2 4 = 0.4 = L 10 3 If we approximate L=0.4 by 3 3 + 3 = 14 2" 0.519 L=0.4 27 S3=0.519 n=3 1 Remainder L - Sn = 0.4-0.519 = - 0.119 4 a -- n+1 = 3+1 a a - 2 3 (2) Error = IL - Snl = = E = 0.119 < Uu = ay ) = (3) 3 =0.198 3 4 3 2 2 = - + 0.321 3 3 3 min {53,54} < L L max {53, 543 min { 0.519, 0.321}
8 7 Exp Approximate the sum of Ety," (2n) with error of magnitude n=o less than 5 x 10 ' 5 x 10 We use Sn to approx. the sum So we need to find n I 6 5x 10 (2n) ! (2n)! > 106 = = (10) (105) = 200,000 5 x10 6 5 5 (2n) > 200,000 n25 I S 1/1 I it 0.54 5 21 41 61 8! Error = 11 Snl = IL - 5 5 5 1 = 0.275x10 -6 6 (10)!
8 10.6 Lecture Problems 14, 19, 29 Cony/Div 8 n+1 14 (-1) 3 n+) Un = I ant n=1 Jn + 1 an Un = 3Vn+1 Vn +1 3Vn+1 3 lim Un lim 3Vn+1 = lim Vn n 8 n 8 Vn +1 n 00 Un+1 un n+1 = lim = 3 n n 0 I + = lim 3 1+ = 3 #0 n 8 I + So the alternating series div by AS nth term test
8 9 E n+l 19 (-1) n n3 + I This is alternating n=1 Un 1. Un >0 2. Un 3. lim Un = lim n = O n 8 n 00 n3 +1 so this series conv. by AST To check if it conv. Abs. 8 8 8 & E n an/ =
10 8 EE" n 29 tann This is alternating n - + 1 n=1 Un If the alternating series Conv. Abs. then it conv. That is if Elanl conu. then {an conv. Abs. 00 00 Elaml = - { tan we use IT n - +1 HA n=1 n=1 cont. +, n 8 on [1,00) If b 2 tan X dx = lim tan'x dx x2 +1 b 8 x2+1 / tan'b u= tan'x = lim u du du = dx b 8 x21 #4 X=1 tan't tan'b = lim u 4 b->00 2 x=b u=tan'b 2 2 = 1 lim 2 (tan'b) - (#) b 8 1/2 [5]2 (2)2 = 3 II 2 32 so { (-1) tann Conv. Abs. = n - +1
Alternating Series
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