SECTION 11.3 THE INTEGRAL TEST AND ESTIMATES OF SUMS ¤ 467 5. The function i ({)= 1 (2{ + 1) 3 is continuous, positive, and decreasing on [1> "), so the Integral Test applies. ] " 1 1 (2{ + 1) 3 g{ = lim w<" ] w 1 1 (2{ + 1) 3 g{ = lim w<" 3 1 4 1 (2{ + 1) 2 w 1 = lim w<" 3 1 4(2w + 1) 2 + 1 36 = 1 36 . Since this improper integral is convergent, the series " S q=1 1 (2q + 1) 3 is also convergent by the Integral Test. 7. i ({)= {h 3{ is continuous and positive on [1> "). i 0 ({)= 3{h 3{ + h 3{ = h 3{ (1 3 {) ? 0 for {A 1, so i is decreasing on [1> "). Thus, the Integral Test applies. U " 1 {h 3{ g{ = lim e<" U e 1 {h 3{ g{ = lim e<" 3{h 3{ 3 h 3{ e 1 [by parts] = lim e<" [3eh 3e 3 h 3e + h 31 + h 31 ]=2@h since lim e<" eh 3e = lim e<" (e@h e ) H = lim e<" (1@h e )=0 and lim e<" h 3e =0. Thus, S " q=1 qh 3q converges. 9. The series " S q=1 1 q 0=85 is a s-series with s =0=85 $ 1, so it diverges by (1). Therefore, the series " S q=1 2 q 0=85 must also diverge, for if it converged, then " S q=1 1 q 0=85 would have to converge [by Theorem 8(i) in Section 11.2]. 11. 1+ 1 8 + 1 27 + 1 64 + 1 125 + ··· = " S q=1 1 q 3 . This is a s-series with s =3 A 1, so it converges by (1). 13. 1+ 1 3 + 1 5 + 1 7 + 1 9 + ··· = " S q=1 1 2q 3 1 . The function i ({)= 1 2{ 3 1 is continuous, positive, and decreasing on [1> "), so the Integral Test applies. ] " 1 1 2{ 3 1 g{ = lim w<" ] w 1 1 2{ 3 1 g{ = lim w<" 1 2 ln |2{ 3 1| w 1 = 1 2 lim w<" (ln(2w 3 1) 3 0) = ", so the series " S q=1 1 2q 3 1 diverges. 15. " S q=1 5 3 2 I q q 3 =5 " S q=1 1 q 3 3 2 " S q=1 1 q 5@2 by Theorem 11.2.8, since " S q=1 1 q 3 and " S q=1 1 q 5@2 both converge by (1) with s =3 A 1 and s = 5 2 A 1 . Thus, " S q=1 5 3 2 I q q 3 converges. 17. The function i ({)= 1 { 2 +4 is continuous, positive, and decreasing on [1> "), so we can apply the Integral Test. ] " 1 1 { 2 +4 g{ = lim w<" ] w 1 1 { 2 +4 g{ = lim w<" 1 2 tan 31 { 2 w 1 = 1 2 lim w<" tan 31 w 2 3 tan 31 1 2 = 1 2 2 3 tan 31 1 2 Therefore, the series " S q=1 1 q 2 +4 converges. 19. " S q=1 ln q q 3 = " S q=2 ln q q 3 since ln 1 1 =0. The function i ({)= ln { { 3 is continuous and positive on [2> "). i 0 ({)= { 3 (1@{) 3 (ln {)(3{ 2 ) ({ 3 ) 2 = { 2 3 3{ 2 ln { { 6 = 1 3 3 ln { { 4 ? 0 C 1 3 3 ln {? 0 C ln {A 1 3 C TX.10 F.
