Question

Part A [15 Points]: Choose TRUE or FALSE for each of the following items. 1. If the series anx" converges, then anx" → as n 700. TRUE FALSE 2. The series & {-1}" is absolutely convergent. TRUE FALSE 3. The series 2 is convergent using the Ratio Test. TRUE FALSE 00 4. The series An- n n2+1 is convergent using the Geometric Series Test. TRUE FALSE 5. The series 2n=1 42+2n+3 (-1)" is conditionally convergent. TRUE FALSE

Answer 1

then Necessary Condition (1) If the series Ean un converges for convergent : wth term of sequin Lan> is going to tero but here anzen oo as no Hence Given statement is False. (2) Given series Σ HD) 깥 Let can>= 1 sequn of positive terms and sequn can> is nk decreasing and Eiman = lim Iris n700 no Hence <any converges to zero By Lebniz test for alternative series Hence sweis E6400 is convergent then series Elton is nk nk absolutely Conuregent Hence Series E40m is absolutely convergent Hence Given statement is true

(3) Given series En! an bey Ratio test Let lim an L n700 an (i) l>I then Series Eam is contigent divergent (ii) LLI then senus Ean is convergent (ii) L=1 then test fail where an=n! an Now lim nti! no ant n! an L=lim nti! .2n n->00 2n2.n! L=lim (n+1) n! noon! 2 L = lim nti no = 0o ( limit does not exits) 2 Hence Series Enl is divergent Series Hence Given Statement FALSE (4) Given series n 2211 n=1 Now It 2 t 2 + 3 + 5 10

Now Me En n n (14/2) n=122 (1+/m2) n=1 nent n=1 ut lant = 1 n(1/2) and <br> = 1 and cany be a positive n svies lim n700 an lim bn non (172) In = lim n -=1 to (finite) no nult) Now n=1 n241 {bn = {Yn is divergent ley P test then series en is divergent Series by limit compouson test Hencu Give Statement is FALSE oo (5) The Series E (in n=1 n21 2013 nditionally, convergent: Let Elanl is convergent then series can is whditionally convergent E Z Ě oo Now (7in n2+2013 227 2nt3 n2 (1+2+3/2) n=1 n=1 n=1 CO 3. (149,4% 1 n=122 (1+2+3/2) by p-series test Convergent, Pyl divergent , PL | El up

Now Where P=271 - then is convergent Series ห=) 22 2. (1+2+3/2) Hence is conditinally convergent mai 12 (1+2+3/m2) E Hence Ginen Statement is TRUE. (AW)