Question

Question # 5. (8 marks) Determine which of the following series converge and which diverge (don't forget to state the tests from class that you are using!). If any of the convergent ones can be evaluated by the techniques we have learned in class, you should give their value as well. An: + 3n" + 7n +2 In(In(n)) (1) V3n13 - 2n° +55 +n +3 An In(n) 8] (ii) ns16 Σ (3n)! (n!) 33" (iv)

Answer 1

() ?N J Hw+3+ 3rt 4 ทฯ [ 30% ท++ n+ 3 an ทฯ ana “ + 3ทไ+10+ สี 1 313 _ an+F+n+ 3 9 S n 1 ++ 3 1 9 ++ h9 1915 1 2 ท 13- 2 + S + 3 ។ 28 13 ท 6/3 4n 1 q+ 3 + J صو ทฯ \\\u 1 3 3 4 5 + | นง { 18 h/3 1ate bna lio Jฯ+ 3+ 3 + lim an lim พฯ * کم کی ทั้ง 13-21 ff 3 ทฯ h8

» lim an lim J५ لم sto no br nya 3 Soj lim an to now bn So by limit Comparision Test is Divergent by as E bn = El b. Test Ean is So, behaviour of also some Series of Ebu So's & W E ana ni Juntan" +70+2 553 m3_2n +5 hitht? Divergent by Limit Comparision Test is

San = Incenimis) Un lnlnl n=16 an= Inchalms) un laims as ; lining an j mal ln (dolni) < halno In doin') LI Anime => 2 Insholm) yndrino ㅗ 9) So by Direct Comparison Test as S Eur Divergent by p-Test So E Anclaimed Unown is also Divergent by Direct Comparision Test Hence. Mo Inceninis Yn enine is Divergent is n=16

♡ مه (3n)! Ti) = 2 an 3 im!°1331 ha ana Bni! 3 In!) 133) ante (3(n+1)! 3 11 n+1)!)3 ( 33.ht, 3 Apply Ratio Test lim - lim (3n+3)! mim!(33 1 (n+1)!23 13319.33 (3101 anti an nyoo 3 no = lim ( 3n+3) (3n+) (3n+1) (m!33 + I (n! 33 nyot 3

Elim (3n+31( 3n+3) (3n+1) 33 inti3 no an :) no 3 a limi (142)(1+y)('++) 27W 17/143.)?. 33 = lim 27 < 97 = nya 33 33 By Test Ratio Ms is Convergent The Series (3n)! (b! 33"

civi bola 6 E ha DEL By p-series Test ; Pri Р then Series Convergent is Convergent So, 3M8 Hence E sa is convergent nel ha

As : Sinua H- u 3 다. HUIS |- 해 + . 3! Sinu a 9. - 1 9 Sinu 2 / t NO + 거 삥 as Sinu 뺐 )- H M H 지 S! On Combo Coefficient ( +- 1 다. + 다 ) 9 기 3! > + + 1+ - مهكر 10 D. 도 이 9 거의 & 을 So 거의 na