Foolig Nwto s Mthod a Fid a formla for th Nwto sqc, ad vrify that it covrgs to a ozro of f. ( si si + cos 4 4 3 4 8 8 bt f. b Fid a formla for f ( ad dtrmi its bhavior as. f ( cos si + as A Stirlig-li Iqality Itgrat th lft ad right sids, potiat, ad complt th iqality: + <! < +. II. Fid th itrval of covrgc of th powr sris So th radis of covrgc is. At, th sris is I. So it divrgs by compariso. +!. a lim lim + a!, bt > >, by Part +! + + 3 + +
At, w gt th altratig sris gt a (!. From th ral Stirlig Iqality, w that a <, so a. Also!! + + + ( + a ( +!!!!. + +! + + Bt + <, so a dcrass to zro. Th Altratig Sris Tst implis that th sris covrgs at. So th itrval of covrgc is,. III. a If is a positiv itgr, fid th radis of covrgc of th powr sris ( ( + ( + ( + (!!. a + + as. So th radis of covrgc is a b If chc th dpoits. I this cas th sris is, which divrgs for ±. At c If, s th rslt of I. to chc th dpoits., w gt th sris (!, bt!
(! > + +! + ( + ( + + > + + ( + From Part I., so it divrgs by compariso. W ca actally s th Ratio tst to dtrmi covrgc at both dpoits: ( ( + ( + ( + a At ±, >, which implis that a+ > a, so lim a, a ad hc lim a ad w gt divrgc at both dpoits. + + Evalatig Propr/Impropr Itgrals with littl or o Itgratio. l I. For th impropr itgral d + Us th sbstittio to fid its val. l l( l d d d l, so w ca cocld that + + + + d. II. Evalat l d sig th sbstittio ( + ( + +. {Hit: z z z.}
l l l d d d, so w ca + + + + + + ( + + + cocld l d. ( + ( + + III. If yo s th sbstittio i th itgral d, yo arriv at d d ( d d. Is it oay to cocld that d? Eplai. Sic th impropr itgral is divrgt, w ca t cocld that its val is zro. IV. a Us th sbstittio alog with th idtitis si cos cos si to valat th dfiit itgral si si( ( ( si d. cos + si cos d d d cos + si cos + si si+ cos ad This implis that si si cos d d + d cos + si cos + si si + cos d
si So d. cos + si 4 b Evalat th dfiit itgral Sam as th prvios problm. ( si ( cos + ( si d for a positiv itgr. V. Evalat si d sig th sbstittio + cos ad cos cos.. ad th idtitis si ( si( ( si d d cos cos + + si si d + cos + cos ( si d + cos So w cocld that d d d ta + + si So d cos. + 4 si
f d f d by showig that VI. Show that if f is cotios th ( si ( si f ( si d sig th sbstittio symmtry. f ( si d f ( si( + d ODD EVEN, si + cos, ad f ( cos d So w ca cocld that th val of th itgral is zro. Limit Problms si I. What happs if yo try L Hopital s Rl o lim +? si cos + si ( + cos + si si + cos si + cos lim dos t ist, so L Hopital dos t apply. II. + si lim. + si + cos
+ cos Agai, lim iqality dos t ist, so L Hopital dos t apply, bt w ca s th dobl + c III. Fid th val of c so that lim 9. c + si +. + c l c c + + c l l c c + c l c lim has th form, c + c c ( + c l c + c ( c So c + c c c lim c + c So lim c b b IV. Fid a simpl formla for lim b b ( + c( c c b, ad hc that, for b >. b ( b l b ( b ( + l c b b b b b b b b b lb b b lb lb b b 9 c l3. ( b lim ; + l b + lb + lb
V. Fid lim si. L Hopital s Rl wo t wor, so try somthig ls. ta si cos si + sc ( ta cos + si Bt lim dos t ist, so L Hopital dos t apply. sc si si ta ta si si lim ad lim t lim ±, ta t ± t So th limit is. VI. Fid th followig limits: l a lim First lt s vrify that L Hopital s rl applis: lim l lim, so lim has th form. l + +
( + + + ( + + lim + + b l lim l + + + VII. Fid lim ( + ( + ( + by obsrvig th followig: ( + ( + ( + l l( + + l( + + + l( + l l ( ( l ( ( l ( ( l + + + + + + l( l ( l( l l l l + + + + + + + + + trms l ( l ( l ( + + + + + + Th last prssio is a Rima sm of som dfiit itgral. l + d l 4
VIII. Th altratig sris dos it covrg to? ( + covrgs by th Altratig Sris Tst, bt what Fid lim + + +, ad yo ll ow th sm of th sris. + + Mthod : Calclat lim + + + + + + by rwritig it as lim + + + ad idtifyig it as a dfiit itgral. + + + d l + Mthod : + + d < + + + < d + + + + l < + + + < l + + + + IX. Tlscoprs a 3 N ( ( + lim ( + ( 3 + + N ( N + N + ( ( N lim + N N + + N b + + + lim N + + + 3 N N + lim N N +
c I. ta ta ta + + + For ( l l l >, l compariso. l ( ( ( ( N ( N lim ta ta ta ta N + + + lim ( ta ( N + ta N 4 Assortd Sris >, so l l l < (, so w hav covrgc by II. For 3 ( l( l l >, l( l compariso. l ( l >, so < l l, so w hav covrgc by ( l( l ( III. a Show that + + + + + + + + +. { } b Show that if { a } is a sqc of positiv mbrs, th if l( { a } is dcrasig. I othr words, show that if l( a l( a Sppos that l( a l( a a is dcrasig, th +, th a + a. +. Sic th atral potial fctio is mooto, l( a+ l( a a a +
c For >, show that Sic l( l +. + dt + t ad + t <, w gt that l( + < dt + d Show that a l is a dcrasig sqc by showig that f l + l has a gativ drivativ. l + l l + + l + < + Dtrmi whthr th altratig sris < < + + + is absoltly covrgt, coditioally covrgt, or divrgt sig th prvios rslts. It is covrgt by th Altratig Sris Tst. Th sris of absolt vals is +, + bt >, so it s ot absoltly covrgt. Thrfor th sris is coditioally covrgt. IV. a Startig with ad gt d, yo gt that! ad fid th sm of a sris. +! d +. Now itgrat from to!. Evalat th itgrals o both sids of th qatio
( +! b Yo ca vrity th sm yo fod i part a by oticig that. So fid th sm ( ( + + ( +! +! +! +! +! of this tlscopic sris ad vrify th prvios rslt. lim + + + (! (! N + +!! 3! ( N +! ( N +! lim N ( N +!