Question

10. Determine if each series is convergent or divergent. Show all your work and state which test(s) you are using. (10 points each) (a) Em=1n\n2+1 (b) X=1 (1+202) (c) Im=2nJinni

Answer 1

(A) Given til Seties integral. Hest If there exists an ♡ ZK So Yar for all nzn , f(n)zan is positive, Continuous and decreasing Then & an and goodx either bosh Converge or diverge : p an : n, integral e compute the indefinite C Capely u-substitution Latted ja to da G-1:-(-u?+1) du drop Citipidu Bocauiti du - du use the common integral -- Con latil In Wu-"))

Sunstitute back as Joshti N 2 2 2 :-( nown as ) Inbul) --(\*b)-f boalan-1) - Chei a l) (padan!) - - In Dwi tyl 3 m) Join 1 - n / Joventil a X lab wpt) -ilte Compute the boundlanes ? Bo ttin din ahmin (as im Jahr 19- 4 105 1-) 21-11 +1 +1 nyit i -In (52 +1) + ln (T2-1) limbo (- kinomat +1] + 6 In | Swi-11) =0 .-In (52+1) + In (12-1) In(52-1) – In [1+7) Converges

- (c) Given & noinn 2 0 Apply series integral Test N3k so meat for all man zN, 6(n) = an is positive Continuous and derveasing of an and Prcx) dx fixer bole Converge ar nik diverge 18on - do J n d nm) Compute integral the indefinite • S muhas on Apply Un Substitution uilnin). full du 91 12 ati a 6 Yeti -V3+ 1 hot Substitute back 4: In(n) In -/+ (m) -Y2+1

L -1+ (2). In = In Wind 2 2 .1n %? (m) Apply radical rule atoa nja Sincn) oz = evinin ! 2) inen) to Compute the boundaries an - J nin(n) lima 2 + (2 Xinen)) »> 2 Din(3) limis (2 sinema) 2. list (lin(n)) e lihao Cincos 12500 - n- D | diverges