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# sol8 - MATH 138 Calculus 2 Fall 2010 Solutions 8 1(03.§a.=...

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Unformatted text preview: MATH 138 Calculus 2 Fall 2010 Solutions # 8 1(03.§a,.= zit—1)"- series converges by the Alternating Series Test. .109)“, Z ":1 m -4 l . . . 2—,: + l =.§1 (-1)" hn- NOW bn = 2n + 1 > 0. {bu} IS decreasmg. and 1111010 bu = 0. so the _°c_n—l 1 _°°_n~l __ 1 _.:2:1( 1) tn(n+4)—,§.( l) b"'N°w b"‘tn(n+4)>° {b"}'5de°'°as'"gra"dn"l‘;b"=°"° the series converges by the Alternating Series Test. A. Le) = % >0f°”‘> 1- {bu} isdecreasingfornz lsince a: '10“(1)—x-10’ln10 10‘(l—mln 10) l—zlnlﬂ —- =———————'=—————= <0f1— l10<0 => '110>l => (10:) (101)2 (10:)2 10: °' 3 n ‘ " l i‘ m n x>m~0.4.Also,n1Lngobn='lgtgo 10" =33; 10’=x‘1l-go-l—0‘lnlozo'Thus’mescﬁcs2H) 1%" converges by the Alternating Series Test. 1M) 2 “n — ZIP 1)" 2n 3124.1: 2(- 1)"b... Now "lLIrgobn- — ”'me 2+ _ 3:: = g- # 0. Since "lingoan 95 () "=1 u=| "_ _: (in fact the limit does not exist). the series diverges by the Test for Divergence. l (a) on 2 2 2 _ . n on“ _ . (11+ 1) 2" __ , l . . The series 2 — has positive terms and lim — lun [ 2" +1 HT? — "111.1; 1 + 12"n-ooo a" n-ooo series is absolutely convergent by the Ratio Test. 09 "+1 m l . - . , ' . . ' s - 2. Cb) 2 {'1‘} converges by the Alternating Series Test. but 2 7.; I33 divergent p-sencs (p = 5 _<_ 1). so the given sum: n "=1 ":1 is conditionally convergent. 1m) °° l converges by comparison with the convergent geometric series "‘4; Z; [M = § < 1] . sin 4n °'° l—4 my“ 0° . Thus. 2 sm4n is absolutely convergent. . el/n 1/" 1 3 > 1] £18 .convcrges, and so e 6 _ cut series itd) . SinceOS-nT SﬁseCn—a )and £1713 isaconverg P' [1): "_ l l‘ f: ( l)"e I . IS absolutely convergent ‘ .,._ "=, n3 . ,-7 f. v 'W I“! - “W“ _ M. L10" = 111__ :1 th W dive s b the Ratio Test. 21(9) "ll-"3° an ‘ .319; [100w T! .3320 100 °°' 3° “ms nZ=:,'—1(1t)v-'*"° y ﬁg) a...“ _ (n+ 1)2 2"+1 n_! _ . l 2 2 _ . m _ "+1 7122" . "lingo“ — "132° [WWI—'12" - “11.11;“ 1 + n --—n + 1 — 0. so the senes "g“ 1) T1! IS absolutely convergent by the Ratio Test. 9 ’ By the recursive deﬁnition, "Ii-u:a “2'“ = "lingo I“?! = 0 < 1. so the series converges absolutely by the Ratio Test. ...
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