Optio Chaptr Ercis. Covrgs to Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Divrgs 8 Divrgs Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Covrgs to Covrgs to 8 Proof Covrgs to π l 8 l a b Divrgt π Divrgt l k 8 a) Ara icrass without boud, i.. ifiit π uits Th ara of th rgio is ifiit; howvr, th volum of th solid cratd b rotatig th rgio about th -ais is fiit. Chaptr Ercis. a) 8 a) ; divrgs b th trm divrgc tst ; covrgs to l l l ; divrgs b th trm divrgc tst ; covrgs to 8 8 8 ; divrgs b th trm divrgc tst ; covrgs to a) d ( ) C d ad thrfor th sris is covrgt. a) Divrgt Covrgt Proof For, lim but it is a p-sris with a p so th sris divrgs. Proof Covrgs 8 Divrgs Covrgs Covrgs Covrgs Divrgs Divrgs Divrgs Divrgs Divrgs Covrgs 8 Divrgs Covrgs Covrgs Divrgs a) S. 8 8; rror < 8 S. 8; rror <. a) ( ) b π d lim [ arcta ( ) ] arcta () ( ) b arcta ; sic d covrgs to ( ) arcta, th must also covrg. Divrgs a). 8 with rror <. trms trms ( ) 8 is coditioall covrgt. Covrgs absolutl Covrgs coditioall Divrgs Covrgs coditioall
Aswrs Covrgs absolutl Covrgs absolutl ; th sum of this sris 8 is. Th trms of th altratig harmoic sris ar rarragd such that coscutiv positiv trms ar addd util th sum is gratr tha, th coscutiv gativ trms ar addd util th sum is lss tha, ad so o. Not that th diffrc btw th partial sums ad is lss tha th last trm usd, so th sris covrgs to. trms Proof Chaptr Ercis. R ; < R ; < < R ; < R ; R ; R ; R ; < < 8 R ; < R ; R ; < R ; < < R ; R ; < < R ; < < k k a) ( ) ;< < A, B ; < < a) ( ) ; R d ( ) ( ) ( ) ; ( ) radius of covrgc is also R. d. ; rror < a. <. 8 a) for < < a) ( ) Proof a) si si π. 8 8 Error <. < < ( ) ( ) ( ) ( ) ( ) a) ( ) Proof Proof d) π. ; rror <. 8 () ( ) a) f ( ) f ( ) f 8. d) Error <.. < <. 8 8 sc a) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d) ( ) a) Practic qustios l ( cos ) a) si cos si sc a) l( ) l ; ( ) R ( ) ( ) ( 8 a) 8
R ( ) ; sic < (. ) th P ( ) ( ), which is a parabola with ( ) vrt (, ). R ( ) (. ) <. Divrgs b compariso with th harmoic sris trms dd;. a) S k S S k S ( ) a) k k k k Proof a) S uk uk u ( k) d k k k k k Error < k k k( k ) Divrgt b limit compariso tst a) ( ) a) Itgral tst for a : Lt a Proof Proof d) π f (),whr f() is a cotiuous, positiv ad dcrasig fuctio for all N ad N is som positiv itgr. Th th sris a) ad a ad th itgral f ( ) d both divrg or both N N covrg. That is, if th itgral is fiit th a is fiit, ad if th itgral is ifiit th a is ifiit. Covrgs b itgral tst 8 a) (i) si (ii) a) si ( ) ( ) ( ) ( ), < < a) Sris covrgs b th ratio tst Sris covrgs b th itgral tst Sris covrgs b th altratig sris tst a) Sris covrgs b th ratio tst Sris divrgs b th itgral tst a) Proof k Proof d). 8 a) ( ). ( ) a) R Covrgs b th altratig sris tst; sum. a) ( ) Proof a) Covrgs b compariso tst Covrgs b altratig sris tst a) Proof (, ) l( si ) Ratio tst givs itrval of covrgc as <. Proof lim si cos lim si si cos si lim cos si Proof cos a) (i) si ; si ; () cos ; ( si ) () si ( si ) ( si ) cos ( si ) (ii) Proof (i) l( si ) l ( si ( ) ) (ii) l(cos ) (iii) ta R a) (i) Covrgs b altratig sris tst (ii) S. 8 8 (iii) Error <..
Aswrs (i) si (ii) a ( ) ( ) (iii) Proof (iv) cos a) 8 a) (i) Proof (ii) Covrs is ot tru; a coutr ampl is which is covrgt but is ot. (i) k > (ii) k a) (i) Proof (ii) a R π. a) lim a. < <. ( ) N, for odd Covrgt b compariso with gomtric sris a) Proof Proof S a) Proof Proof GDC valu: l (. ). 8 For. :.. For. :.. Th scod approimatio is arr to th tru valu. α a) (i) I l (ii) lim I l ( α ) or lα a) (i) (ii) Th argumt is icorrct bcaus th domiator is ot zro wh. a, b, c, d 8 a) Covrgt b compariso tst, or limit compariso tst (i) (ii) ( ) < a) 8 a) A, B S r r 8r lim ; hc, th sris is covrgt. p > a) l () ( ) a) (i) f ( ) (ii) f( ) (iii) f 8 8 8 (iv) Error <. Sris is covrgt b itgral tst a) (i) Domai [, ], rag π, π (ii) arcsi cos(arcsi ) 8 r q (i) p p p r q r r p r r q p ( ) (ii) p, q, r ; hc, th sris i ad is ( ) sic cos arcsi cos arccos. ( ) ( ) ( ) a) Proof a) (i) Proof (ii) Proof (iii) p, q Proof Proof 8 Sris is covrgt b th ratio tst a) Proof 8. < <. a) f ( ), f ( ), f ( ) ( ) ( ) l 8 d) Error <. ) Actual rror l. 8; uppr boud for rror foud i d) is much largr tha th actual rror, so th stimat of foud i caot b cosidrd a good stimat. a) π a) Proof Usig π : l π π 8 ; usig π : π π l. a) R Covrgs b limit compariso tst (comparig to covrgt sris ) a) Proof l ( si ) l ( si )
d) Proof ) 8 Giv a ad a f (), itgral tst ca b applid if f() is positiv, cotiuous ad dcrasig for > ; covrgs. 8 ( ) ( ) ( ) ( ) 8 a) for si π d) ) f) a) Covrgs; gomtric sris with r, so r <.. Divrgs b th trm divrgc tst Covrgs b limit compariso tst d) Divrgs b itgral tst ) Covrgs; compariso tst (compar to p-sris with p ) π ( ) π ( ) a) si ( π). Chaptr Ercis. (i) c (ii) a (iii) d (iv) b a) C C l ( ) l C or C d) C si or arcsi C ( ) ) C f) C g) l C h) l C d d d d l l C l l C l l C A ± si C Th costat C caot b compltl arbitrar bcaus si C. If C <, th si C will alwas b gativ, rgardlss of th valu of. If C >, th si C will alwas b positiv. If C, th whthr si C is positiv or gativ will dpd o th valu of. a) d) Rgardlss of th iitial valu of th populatio, as tim icrass, th populatio stabilizs at. ta a) Proof 8 (i) b (ii) d (iii) c (iv) a a) ( ) ( ) proof a) C( ) d ; d itgratig factor is ; lads to sam solutio as i part a) C a) C cos C d) cos C ) C f) l C csc C csc a) Proof arcsi d) a) Proof ta C sc C l 8 C ( ) a) C C C C d) C C ) f) l ( C)
Aswrs a) Proof Proof Proof a) C Proof a) 8. (i) (ii).......8.8. d) appro. act % rror..........8...8..... 8 C...... 8.... 8 Sic d is dcrasig, th valu of is ovrstimatd at d ach stp. a) a, b, c (i) I l l arcta k (ii) C π l 8. a) Proof sc si cos a) Proof l C l. 8 a) Proof a) Proof si si a) si 8 a) Proof (). 8 (). 8 8 d) Actual valu to s.f. is (). 8 ; usig mor stps (ad a smallr stp siz) givs a bttr approimatio. Practic qustios a) ( ( ) C C si cos C a) a) d) ) f) 8si 8 8 a) ta sc a) Proof ( or ) a) (i) Proof (ii) (. ). (. ). d) Approimatio usig Maclauri sris ca b mad mor accurat b computig mor trms of th sris; approimatio usig Eulr s mthod ca b mad mor accurat b dcrasig th stp valu.
( ) a) Proof l arcta l C a) arcta l a) (i) (. ). (ii) Dcras th stp siz l ( ) C 8 ( ) C( ) R E L I R R E ; I R E L t R sc ( ) l ( cos ) C