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# Class Note for MATH 1432 at UH

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This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15

11 Basic Properties In nite Series De nition Given a sequence akz 1 the in nite series is de ned by the limit of the se quence of partial sums 00 n E ak lirn E ak TL gtOO k1 k1 The series converges if the limit exists otherwise it diverges 1 3 7 15 31 E Z 5 E 5 1 I III 31 5392 53 S4 55 ilil1111 1 H 2k 1 2 4 s 16 32 1 Remarks The summation index is a dummy index7 much like the integration variable in a integral 0 1 0 1 00 1 Z 27 X g Z 27 kil i1 711 The starting value of the surhrnation index needn t be 1 and may be chosen for convenience 00 1 0 1 0 1 27Zgi1ZQn 1 i0 1 k1 i 000 1 7122 1 1 i1 k0k1k2ii1ii1Ti1 i i1 1 11 14r 1 1 1 1 1 sn gt asn gtoo 1 2 2 3 i i1 i1 Basic Test for Convergence 00 If Zak converges7 then ak gt0askz gtoo k1 00 If ak we 0 as k gt 007 then 2 ak diverges Emample 1 k1 k 1 y 0 k th 0 k 1 2 d39 gt as gt00 en 1eres k 1 H k 1 2 3 V g Remark There are divergent series for Which ak gt 0 1 00 1 1 1 E gt0asl gtOO but ggl diverges Basic Properties 00 oo 0 If E ak converges and c is a constant then E cak converges and k1 k1 oo 00 E cakcE aki 161 k 1 0 1f Zak and Zbk both converge then 2ak bk converges and k1 k1 k1 Z bk Zak Ebb k1 k1 k1 2aak bk aZak Zbk7 V0176 E R k1 k1 k1 Quiz Quiz 1 3 ldz or a 1 b 2 c 001 4 ldz 1 z a 1 b 2 c 001 1 2 Geometric Series Geometric Series Geometric Series 2201 00 7 21k 171 k0 diverges if 2 1 k if lt1 Proof 0 We have a closed formula for the nth partial sum 1 7 zn1 171 7 1f lt 1 then zn1 A 0 as n A 00 thus 3 A sn1z12z 1 L 171 If gt 1 then zn1 is unbounded thus 3 divergesi 01f11thensnn1gtooasngtoo 0 If I 71 then 3 alternates between 0 and 1 thus divergesi Examp m 15 31 16 32 N in Nlt gtw if COII 1 III S4 55 0 1 1 1 1 0 1 1 E32 248 27 11 00 k 1k 1 1 1 lt1 1 2 k 1 0 0 1 1 1 2 201111111mzz 1 k k 1 k110 130 1E 9 iii 09999999 900i 9gtltl 1 10k 10 9 k1 k1 00 1k gtakE09W0a1a2a3a4a5 1n n1tedee1mal Examples Suppose that a ball dropped from a height 11 hits the oor and rebounds to a height 011 with 0 lt 1 and so on Find the total distance traveled by the ball if h is 6 feet and 0 The total distance traveled by the ball is D h20h202h203hm m h20h1a02h20hak k0 1 2 1 h20h 62gtlt7gtlt6 236feet 170 3 17E Examples 00 m m 3H1 7 2k 7 23 73 3H1 2k 72 7 i 7 if i 5k 525125 25k 2516 160 k0 k0 00 k 00 k 3 2 1 1 35 3ltgtltgt v k0 5 k0 5 1 3 1 3 6 m m 1 1 1 1 1 1 iii i 7 iil Z3k 2781243 Z3k ltg9gt k3 k0 7 1 1371 717 9 18 m m m 1 1 1 1 1 1 1 3 g Zgws Zgik k3 10 k0 i 1 i 2717 18 Outline Contents 1 In nite Series 111 Basic Properties 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 112 Geometric Series Lecture 21ection 111 In nite Series JivvenI1e 1 In nite Series What is an In nite eriFsZ 1 1 Let the sequence ak 27 5 I g E 1 Form the partzal sums 1 1 31a1ak k1 2 1 1 3 SQialHQigakg le 1 1 1 7 837a1a2agigak47 l g 7 Z 1111 15 sa a a a a7 7 7 ii 12341621162481616 n 1 1 1 1 1 1 7 7 7 7 i1 371 gut 24816 2n 27M 00 n 1 swzawnlggo ezak213200me1 k1 k1 The in nite sum77 21 mm is called in nite series Quiz Quiz a 17 b 2 C 001 a 17 b 2 C 001

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