# 14th - 1 1.3 THE IN'l‘I‘ILRAI TEST 1 converges a...

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Unformatted text preview: 1 1.3 THE IN'l‘I‘ILRAI. TEST 1. converges; a geometric series with r = 11—0 < l x. 7:: ' . . 2 _ o 1 - - , - _ 3 o. converges, E 1 WE _ —_ “E l n” 3 , which Is a convergent p—series (p — E) \- .n u = x — 1 - 11—1 16. diverges by the Integia] Test: 11 W; |: d“: J —. j: ‘13—” = ln (x/ﬁ— 1) — ln 2 \u —- 003511—- 00 . . . 1 . tanfl] . 596“) 24. diverges by the nth—Tei'in Test fordivergence; liin _ ntan (—j = liin _ I D = lim _ ~ ii—ioe rl 11—':')<_- F] ii—ioc- _;U _ - 2 1_ _ 2 _ - —]1]_]!an sec (DJ—sec 0—1;0 l 1.4 C.( )1“ PA RIS( )N TESTS 5. diverges since 11 311221 : i 0 l2. converges by the Limit Comparison Test (part 2) when compared with , a convergent p—series: ri-J [1n niir 'l | . L .- 1 3(1 1' - _ 2 _ 2H 1 E _ li]n_ ["T J : lim_ “Uni : lim_ LE=3 ilm_ (inn) :3 l]!'.ll_ la=15 llﬂ'l_1n'—n 11—‘o<_- n—ux- n n—‘oe 1 n—ioe n li—‘DL- 1 li—‘DL- n = 6 - 0 = 0 I) 19. converges by the Direct Comparison Test with , the nth term of a convergent p—series: n“ — l b 11 for 22 .3 32-3'31-1-" ' '1 11232 2:» n [n —1]>n 2:» n\,n —1>n’ 2:» > 73—101useLimitComparisonTestWithFL 0" ' in n— — 25. diverges by the Limit Comparison Test (part 1) with the nth term of the divergent harmonic series: ' | . \$10 — . ' 1]]‘11_—“—( . )2 lim M21 um (31 Ho x i i 6 ( 6 - , - 36. 1 ’2: r 3.3} } n3 _ mum”) _ “in } “(2M H n3 _.» the series converges by the Direct _5_ Comparison Test 11.5 THE RATIO AND ROOT TESTS . . . . . r 1 i u . 3. diverges by the Ratio Test: liin 3"“ = lim n : liin ‘“ i“) - 3—, : liin “' 1 = 00 h 11 —‘ no 2.: n —‘ oc- 11 —‘ oc- e n. n —‘ oc- e ﬂ—‘ D -1 10. diverges; 11]_iin an : lim (1— = lim I. (1— ( 113]) : e‘hl‘u‘ a: 0.72 ;’ 0 , - . - a.“ _ - tni 112mm i 2:! Fri! 20. converges by the Ratio Test. 1111mm a" _ nli‘mxl - _ - iii 1 2 i112 _ g _]1]_]in3<_-.( ri (mill—3 <1 as '— 3" {5/3 i I . _ . an“ _ . _ . _ _ 32. converges by the Ratio Test. “11in” a" _ 011mm an _ nll‘mxl n — 3 «ti 1 1-3.--t2n— n _ 1-2-3-4---t2n— non) _ (2n!!! 44. converges bytlle Ratio Test: an 2 EHIHBMUH } 1} _ E2_4m2n]._,(3n } 1} _ (znnnjtgu r l} - [211 t 2)! _ g2"n!‘:1§3“ t 1‘.I_ - (En! llt2n t 2);? t 1‘: 2:, n [Zn—1m} 1:]!I'1(_'J.n+l } 1;, [211)! _ n 22in} Dian“ l l} (1 r 3'“) _- 4n'3tsni2 _ _1__ _11]linx.(4n'—t3nl4)(3r3'“}_l 3— <1 .nlg 111v AL'l‘I‘IRNA'l‘ING SERIES, ABSOLUTE AND CONDITIONAL CONVERGENCE TI: TI: ” converges absolutely 2% converges by the Absolute Convergence Test since Z |an| : Z: which is a 11—] n—l convergent p-series 6. converges by the Alternating Series Test since ﬁx) 2 “17" 2:» f’(x) = 111,“ < 0 when x b e 2:» ﬁx) is l decreasing 2:» nn 3-“- nm];also url 2“ Oforn 2“ l and lim uJ1 = lim m—“ = lim = 0 n—‘oc- n—‘oc- 1'1 11—inc- V. k l l. converges absolutely since 2 Ianl = Z n a convergent geometric series n:l 11:1 . x _ ,3 b 25. converges absolutely by the Integral Test since 11 [tan—1 x] dx = lim [0“: x" ] 1 b_.3<_-. [( J we] I) tG|—' tal‘a = lim [nan-1b]2 — [tan—l If] 2 b—‘oc- 46. |error| < (—1)“(%] 20.00001 ...
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