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08.07 - 8.7 POWER SERIES n 1 “n 1 X...

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Unformatted text preview: 8.7 POWER SERIES n+1 “n+1 X n <1=>le<1=> ml<x<1; whenx=m1wehaveE(—1)“, adivergent rt=l 1. lim <1 => lim nwoo nwoo 00 series; when x m 1 we have 2 l, a divergent series n=E (a) the radius is 1; the interval of convergence is —1 < x < 1 (b) the interval of absolute convergence is —1 < x < 1 (c) there are no values for which the series converges conditionally when“ _ n “.100 n+1 (3x—2)" )<1 E} l3X— 2l<1 n+1 <1 => E3x—21nlimm(“ 00 2> —1<3x—2<1z>§<x<1;when)rm§wehave n=1 (T"w which is the alternating harmonic series and IS 00 conditionally convergent; when x = 1 we have 2 % , the divergent harmonic series n=l (a) the radius is %; the interval of convergence is % g x < 1 (b) the interval of absolute convergence is § < x < 1 (c) the series converges conditionally at x 2 % u_a+ l “n <1 =>nlin1 n—+oo 7. lirn00 n—ﬁm (n+1)xn+1 (_n+2)‘ (n+1)(n+2) (n+3) <1 => lxl nlimm (n+3)(n)' <1 é lxl<1 =>—1 < x < 1; when x— - —1 we have 2 (—1)“n“ —2+ , a divergent series by the nth-term Test; when x— - 1 we n=l 00 have 2 n j: 2 , a divergent series 11:? (a) the radius is l; the interval of convergence is —1 < x < 1 (b) the interval of absolute convergence is —1 < x < l (c) there are no values for which the series converges conditionally <1=> fllg'nDO 9. lim n—roo xn-H n_\/'3n 1x_§(nlim _)( (n+1)7n+13'a+1 x—“ <1: rim—mo n—+1 91.190011“ “n1 3"; (l)(1) < 1 => [xi < 3 :> —3 < x < 3; whenx~ — —3 we have 2 \$23311 absolutely convergent series; n=E w when x m 3 we have 2 na—lﬁ, a convergent p—series n=l (a) the radius is 3; the interval of convergence is —3 S x S 3 (b) the interval of absolute convergence is —3 5 x S 3 (c) there are no values for which the series converges conditionally x2n+3 11! 13. lim “gm <1=> nlirn oo (n+1)!'x+ <1 =>X2nlimm(5-11;T)<lforallx “111 (a) the radius is 00; the series converges for all x (b) the series converges absolutely for all x (c) there are no values for which the series converges conditionally 19. 25. 29. 33. 13 - <1=s new “5+1 [1 - n+1xrﬁri 37:: <1 => nlgnm |V 3n+l nlg];00 :1)<1=>3;‘—’<1=>[xf<3 => —3 < x < 3; when x = —3 we have 2 (—1)“\/r_1 , a divergent series; when x = 3 we have 2 \/_, a divergent series “=1 n=l (a) the radius is 3; the interval of convergence is —3 < x < 3 (b) the interval of absolute convergence is —3 < x < 3 (c) there are no values for which the series converges conditionally ' U51 (xi-2W" _n_2” Ix+2i - u §_x+2§ nlgnoo 15—: (13111-131100 W (—_x+2)" <1=> 2 nl—lsmoo (n+1)‘<1:> <1:>ix+2‘<2 W 1 => —2<x+2<2 ¢ —4<x<0;whenx=—4wehaveZ;—Hadivergentseries;whenx=Owehave2%, n=i n=1 the alternating harmonic series which converges conditionally (a) the radius is 2; the interval of convergence is —4 < x S G (b) the interval of absolute convergence is —4 < x < 0 (c) the series converges conditionally at x m 0 1. <1 2} 11 (4x__w5)2“”._2_rn3’2 <1 => (4 —5)2 '1 )3/2<1 => (4 —5)2<1 n—qnoo 11... 11—51100 (“+13312 (4x—5)"+ x nqmoo n+1 X 2:11 °° . . 2> [4x—5f<1 z> —1<4x—5<12>l<x<%;whenx—lwehavez %- ”771; whichls n=E 2n I absolutely convergent; when x— - 2 we have Zn (1) T; , a convergent p—series n=1 (a) the radius is ‘13—; the interval of convergence is 1 S x S g (b) the interval of absolute convergence is 1 S x 5 g (c) there are no values for which the series converges conditionally limm “—3“ <1 => limm #-%§|<1=> “g”? limm ilf<1 => (x—1)2<4 => [x—1§<2 :>—2<x— 1 <2 :> —1<x<3;atx=—1wehave2“—fznz % = Z 1,whichdiverges;atx=3 nﬁ nw we have 2 25:) = mg»; 2 Z 1, a divergent series; the interval of convergence is —1 < x < 3; the series n—0 n—0 f: mun?" n=0 21f)“ is a convergent geometric series when —1 < x < 3 and the sum is I...‘ ﬂ: W— Wmmmﬁ 1‘(T) l 4 i T ...
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