Question

9. [7 points] Use a Direct Comparison Test to determine if the series converges or diverges. 00 7n 4n - 2 7121

Answer 1

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Ans Direct Comparison Test : Suppose two series Zan and E by are such that ang bnyo for alln bu san for all n. and if Then E bn is also ME) (i) If an is convergent then convergent, (33) If a hy on is divergent then & an is also divergent. 12) Now let an 7 - كم in yn

ч" 7) ЛЈою " 70 20 м) — 3 and чи м7) 2) –72 20 17 ) 2) ч" 4 7° 2% м7 ) 7o 263 2 1 3) &0) I because so 2. у" } hal 2 a70, 270 2) a70 — a, 7, о 25 м2 ) - (ba0) similarly 7° 5% 77/ Э by > D 294 171 uh

Now (4u zal uh > Enh (и то -) — u uh ч"- 2 4 ikust uh а) 7 7 ч" Тесалле -">0 24 м, ) and if bcc and - ab cac azO а) bn Lan (24 өх (7. (by D&Q) у Now & but 1 7 Y) на) у

11 SW IT n-1 O - Å bn n-1 کے n=1 m=) -) HD it is geometric series of the form & arnt which is convergent for lrk , divergent for r/> now comparing SOE) n=1 and int with agent, n=1 a= 7 Y - 1 Y

Now 141= 1 / 1 => 1 = I> so E bn n=1 (515) / is divergent Now bay 6, 6.0 bny, o and an7o by <an for nyl Also bn is divergent. result (1) (1) of Direct comparison Test, So by an= { is divergent, NE) n=

given series is divergent