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# 08.06 - 8.6 ALTERNATING SERIES ABSOLUTE AND CONDITIONAL...

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Unformatted text preview: 8.6 ALTERNATING SERIES, ABSOLUTE AND CONDITIONAL CONVERGENCE 1. ll. 15. 18. 19. 23. 29. 35. A? 00 00 converges absolutely => converges by the Absolute Convergence Test since 2 laui = Z [1% which is a n=1 El=l PO“ VPP'UPI'I" ﬂ—QPT; PC converges by the Alternating Series Test because f(x) : In x 1is an increasing function of )1 => ﬁ is decreasing :> uu 2 un+1forn 31;alsoun 2 Oforn 2 landnl_i1+'nm11T-n :0 ﬁ+1 3 f’(x)- i—x—zﬁ converges by the Alternating Series Test since f(x) = _ m < 0 2) f(x) is decreasing => ull zummlsoun 20f0rn 2 landnliin00 11l1 =nlim V5“ :0 90 DC! . I] u - converges absolutely Since 2 13“] = 2 (11—0) a convergent geometric senes nzi n=1 DO 00 converges absolutely since 2 Ian; 2 Z nil—3U and ﬂ < “—1; which is the nth-term of a converging p—series n=l n=l converges absolutely because the series 2 liﬂg— “l converges by the Direct Comparison Test since (Si—n;— ""t S “—1; n=1 diverges by the nth-Term Test since n 131100 ii: = 1 ;£ 0 1 2 Z n+1 converges absolutely by the Ratio Test: nlim00 ( “ﬁg—)2“ “100%;— — g < 1 converges absolutely by the Ratio Test: nleOO “—"j‘)— _ n111-1100 (ﬁg—)3: - (—1%)n = “Ilium “1201 = 0 < 1 l/n— . - 11 (11+ _1)“ - 11+ 1 __ l converges absolutely by the Root Test. 3 111nm:n _. 11100 (—2“, —nl_1111m 2n — 2 < 1 nonveroes absolutely! hv the Direct (‘nmnarimn Test sinnmspnh (n) m 2 _ A ( 25—“ F. — zwbichs thra ...
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