Question

Problem 3: 115 pts Short Answer, Part 1 Directions Answer the questions below. No justification is necessary A. 14 pts Suppose that converges, and define and Complete the table below which shows terms in the sequences fosseland (room. You do not need to provide justification 2 3 8 1 2 -1 B. 16 pts Suppose that ) (-2) and it is known that (0) converges and diverges. Complete each statement below. No justification is necessary! . (1.5 ptal The series O converge O could converge or diverge diverges • 11.5 p) The wries represented by $(7): O convergen, O could converge or diverge Odiverges . 3 pts) The minimum possible interval of convergence for (a) C. 5 pts Give an example of a series that converges as a consequence of the integral test, but for which the ratio test is inconclusive or explain why no such series exists. If you provide an example, make sure you explain why the integral test is conclusive but the ratio test is not

Answer 1

SO! Given table. (A) n - 9 3 an 3 Sn 1 2 ما m 3 -1 S, -21 = 1 Sqaa, tag=2 s-S2 V2 = 2 7 sa S2 +82 - 4 1, -S-S = 4.-13 r3=S-S3 => -1 = 4-63 =) S3=5 S3 = S2 taz 35=2+93 (B) Groan given statement 460) - EP ak(-2)* -> converge o ak gk 3 diverge and K20 K20

ak; By observing both function, we wn conclude that convergence and divergence are ony depend on base of * [C-9jks base: -2] it base 's posibve then dierge and when base is negative then converge. The senes & It could be conwerge and diverge both because use wn not take conclusion about ak from about Statement >> +17): E ak (7-2)* = 40 ak csjk The base of £ ?S 5 (posiblue). Herce ?t is diverge 2) minimum possible internal of convergence for frases When a 2,20 Then a£g minimum possible value of als a E (W, 2] C) for this type of statement we are taking Kro example. E Kol k2 To verity this enampie, we need to perform ratio test and Potegral test. - The nato test states that . anti - an L. im no ait Lal, then convergence it Ly' then divergent It L=1, then test is inconclusive 1

- For given sepies - (Rabo test ) L=1?m k yoo CK12 k2 1?m koo 1 1x2+2x+1 L2 2-18 kyo 1+ 2+ L-1 Inconclusive - Integral test- T S dk 1 k2 Ico[], 12) I-2 (Bounded) Hence és ? convergence accor cording to integrar test but Pt is concuspai according to grato test. k2 KE