10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n

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1 0. Sere I th ecto, we wll troduce ere tht wll be dcug for the ret of th chpter. Wht ere? If we dd ll term of equece, we get whch clled fte ere ( or jut ere) d deoted, for hort, by the ymbol or Doe t mke ee to tlk bout the um of ftely my term? We c proce the followg wy. We coder the prtl um: Thee prtl um form ew equece. We defe If the equece coverget d lm lm coverget d lm ( fte umber), the the ere or The umber clled the um of the ere. Otherwe, the ere clled dverget. We c oly fd the um of very pecl ere. We wll look t how to fd um for Geometrc ere d Telecopg ere.

2 Reult : The Geometrc ere: r r r 0 + coverget f r d t um frt term r r + rto r If r, the geometrc ere dverget. Proof. Defe prtl um So, r r + r r r r r r +, thu we hve r r r + r ( r r r + r ) ( r) r r r From lt ecto, we kow lm r 0 - r, o lm lm( r ) r r lm r doe ot ext r,d r, o 0 r dverget r d r. Thu, whe - r, r coverget d r 0 0 r A r r r r. +, o lm doe ot ext. (ume 0). Thu, o 0 r dverget r. Exmple : Fd the um of the geometrc ere Soluto: the frt term 5 d the rto coverget d t um r ce 0 5 r, o the ere

3 r Exmple : Determe f the followg ere coverge or dverge. If they coverge gve the vlue of the ere.. b. c. 0 0 ( ) ( ) 0 Soluto:. ( ) = ( ) ( ) = ( ), the rto r, r, o the ere coverget. Frt term ( ). Thu, 0 ( ) = ( ) b. ( ) ( ) ( ) ( ) ( ) ( ) ( ) o the rto r, o the ere coverget. Frt term ( ). Thu, ( ) ( ) c ce rto r, o dverget. 0 0 Exmple : Wrte the umber.07 rto of teger. Soluto:

4 We kow tht fter the frt term, we hve geometrc ere: wth frt term, d rto r, Now, we coder other pecl ere clled telecopg ere: ere whoe prtl um evetully oly hve fxed umber of term fter ccellto. Exmple 4: Determe the ere coverget or dverget. If t coverget, fd t um. ( ) Soluto: We c oly deped o the defto of coverget ere d compute the prtl um. ( ) 4 ( ) We c ue the prtl frcto decompoto to get ( ) A B Let, the you c fd out A d B ug the procedure from prtl decompoto ( ) Thu, we hve ( ) 4 =- d o lm lm- Therefore, the gve ere coverget d ( )

5 Queto: Determe the ere coverget or dverget. If t coverge, fd t um. 4 Soluto: 4 ( )( ) Let A B ( )( ), the you c fd out A, d B. Thu, lm lm Reult : Hrmoc ere dverget. Proof. 4

6 5 6 Smlrly, we hve, 64 d geerl we hve. Thu, we hve lm d o dverget. Therefore, the hrmoc ere dverge. Queto: Determe the followg ere coverget or dverget.. b. Remrk: If 0 (dverget ce 0 0 ) (dverget ce coverget (dverget), the M K ) coverget (dverget). Reult : If the ere coverget, the lm 0 Proof. Let 4. The. Sce equece coverget. Let lm. We lo hve lm. Therefore Remrk : If the ere lm lm( ) lm lm 0 equece 0, or lm 0. coverget, the the lmt equece coverget, the or lm, the lmt of Remrk : The covere of the bove reult ot true geerl. If we hve lm 0, we c t coclude tht coverget. For ere, lm 0, but the ere dverget. Of coure, the equvlet ttemet of the bove reult lwy true, whch the followg. Reult 4. Dvergece Tet: If lm 0, the dverget. Remrk: If we fd lm 0, the we kow tht kow othg bout the covergece or dvergece of the ere dverge. dverget, but f we fd lm 0, we. The ere c coverge or

7 Exmple 5: Determe the ere coverget or dverget. 4 Soluto: Sce lm lm 0, by Dvergece Tet, the ere dverget. 4 4 Queto: Determe the ere Queto: Determe the ere rct coverget or dverget. (wer: dverget) l coverget or dverget. [Plee jutfy my followg cocluo correct or ot. Sce lm l lm l l 0, o l coverget. Th ot correct, we c t y ythg bout the covergece or dvergece of the ere l f lm l 0 ] Soluto: Let l l [l( ) l ] (l l) (l l ) (l 4 l ) [l( ) l( )] [l() l( )] l() lm lml(), thu the ere dverget by the defto of dvergece of ere. Reult 5. If the ere, b re coverget, the we hve ( b ) b, c c Remrk: If the ere, b re coverget, geerl, we DO NOT kow whether ( b ) coverget, eve though ( b) coverget for ome ce, geerl ( b ) b

8 Exmple 6: Fd the um of the ere Soluto: ( ) 6 ( ) ( ) 6 6 ( ) ( ) 6 ( ) (frt two ere re geometrc ere,we c ue the formul) ( ) (ue the reult from Exmple 4, we hve ) ( ) 4 5 4