SECTION 85 POWER SERIES 85 POWER SERIES A Click here for aswers S Click here for solutios 9 Fid the radius of covergece ad iterval of covergece of the series 0 3 x 4 0 5 x 6 7 3 x 8 0 9 0 l x x x s 3 5 x x x 0 0 x! x 3 3 3 s 3x 4 x 5 6 s 7 3 x 8 0 s 9 0 0 x 3 5 x 0 x 4 x 4 5 x 3 Copyright 03, Cegage Learig All rights reserved
SECTION 85 POWER SERIES 85 ANSWERS E Click here for exercises S Click here for solutios, [,, (, ] 3, (, ] 4, (, 5 5 5 5, (, 6, [, ] 7, [, ] 8 0, ( 0, 0 9, [, 0, (,, [ 5, 7, [, 0] 3 3, (, 3 4, ( 3, 5 5, (0, ] 6 5, [, 9 7, (, ] 4 8, [0, ] 9, (, Copyright 03, Cegage Learig All rights reserved
SECTION 85 POWER SERIES 3 85 SOLUTIONS E Click here for exercises Copyright 03, Cegage Learig All rights reserved R stads for radius of covergece ad I stads for iterval of covergece i this sectio If x +,the a+ x+ +3 + x + x +3 x < for covergece (by the Ratio Test So R Whex, the series is which diverges (Itegral Test or + ( Compariso Test, ad whe x,itis + which coverges (Alteratig Series Test, so I [, If ( x 3,the ( a+ /3 x x < for + covergece (by the Ratio Test, ad R Whex, ( 3 which is a coverget alteratig series, but whe x, which is a /3 diverget p-series (p <, so I (, ] 3 3 If ( x,the a+ x /[ + ( + +] x /( x + x < for covergece, so x < ad R Whex, ( x ( which coverges by the Alteratig Series Test Whe x, ( x which diverges (harmoic series, so I (, ] 4 If 5 x,the + 5 x + 5 x < for covergece (by the Ratio Test, so R Ifx ±, 5 5 as,so diverges by the Test for Divergece ad I ( 5, 5 5 If x,the + ( +x+ x + x x < for covergece (by the Ratio Test So R Whe x or, x does ot exist, so x diverges for x ± SoI (, 6 If x the a+ ( x x < for + covergece (by the Ratio Test, so R If x ±,, which coverges (p >, so I [, ] 7 If 3 x ( +,the + 3+ x + ( + ( + 3 x ( + 3 x 3 x < + for covergece, so x < ad R Whe x, 3 x ( + ( +, which is a coverget p-series (p > Whex 3, 3 x ( + (, which coverges by the ( + Alteratig Series Test, so I [ 3, 3 ] 8 If x 0,the x ( + 0 a+ x 0 < for covergece (by the Ratio Test, so R 0 If x ±0, as,so diverges (Test for Divergece ad I ( 0, 0
4 SECTION 85 POWER SERIES Copyright 03, Cegage Learig All rights reserved 9 If x l,the a+ x + l ( + l x l x l ( + so R Whe x, diverges because harmoic series Whe x, H x x l l, which l > ad is the diverget x l which coverges by the Alteratig Series Test So I [, ( l, 0 If ( x,the (! a+ x 0< for all x By ( + the Ratio Test the series coverges for all x, sor ad I (, If (x 3,the +3 a+ + (x 3 + +3 +4 (x 3 +3 x 3 x 3 < +4 for covergece, or x 3 < 5 <x< 7,ad R Whe x 5, (x 3 ( +3 +3, which coverges by the Alteratig Series Test Whe x 7, (x 3, similar to the +3 +3 harmoic series, which diverges So I [ 5, 7 (x + If ( +,the a+ x + x + < for + covergece, or <x<0 ad R If x or 0, the + <,so does (p >, ad I [, 0] coverges sice 3 If (3x +,the a+ + (3x + + (3x + + (3x + 3x + as so for covergece, 3x + < x + < so 3 3 R ad <x< Ifx, the series becomes 3 3 ( which is diverget by the Test for Divergece If x 3,theseriesis which is also diverget by the Test for Divergece So I (, 3 4 If 4 (x,the a+ ( +(x + 4 4 + (x x ( + x as 4 For covergece, x < x < R ad <x < 3 <x< 5 If x 3, the series becomes 4 ( 4 ( which is diverget by the Test for Divergece If x 5,the series is 4 4, also diverget by the Test for Divergece So I ( 3, 5 5 If ( (x,the a + (x + + (x x x < + for covergece, or 0 <x<, adr Whex 0, ( (x which is a diverget p-series (p < Whe x,theseriesis which coverges by the Alteratig Series Test So I (0, ] ( (x 4 6 If the 5 a+ x 4 x 4 < for 5 + 5 covergece, or < x < 9 ad R 5 Whex 9, which diverges (harmoic series, ad whe ( x, which coverges by the Alteratig Series Test, so I [, 9
SECTION 85 POWER SERIES 5 7 If ( 3 (x + a+ the 3 x 3 x < ( / + + for covergece, or 3 <x< 4 3 ad R 3 Whex 4 3, ( which is a coverget alteratig + series, ad whe x, a 3 which is a + diverget p-series (p <, so I ( 3, 4 3 ] (x 8 If,the 3 ( a+ 3 x x < + for covergece, so x < 0 <x<, ad R Theseries (x coverges both for x 0 3 ad x (i the first case because of the Alteratig Series Test ad i the secod case because we get a p-series with p 3> So I [0, ] x 9 If 3 5 (,the + + x 0for all x Sothe ( + series coverges for all x R ad I (, Copyright 03, Cegage Learig All rights reserved