University:
California State University, Northridge
Course:
MATH 150B | Calculus II
Academic year:

2023
Views:

438

Pages:

8
Author:

Alifah C.

Integral Test (Conv/Div, IT?8anConsideran/ wheren=1n=Kv.an positive terms andin f(n) is cont. , +1on [K, o)in8Then Ean 8 an and Sf(x)dx both Conv. orn=KKboth divExp Does the following Series Conv. / Div ?&. an= 1/2 positive term I n=1,2,31E. f(x)= 1/Xis +, cont., on [1,00)nn=188dx =\S=1=1Hence,E1/2con to /n2=2J-1n=1Iby ITDIT2EConbyn3n=1conv~r 0 1 1 conv&Exp+dxX XP =P-1 if p>1I8if P I IDivcomExpdr =if pasx2 +1b-soI= limtan b - tan'i]17/47tan'bb-ow= H2 - IF/bI'll-2II=4Hence,Conv by ITn°+1n=1is postive for all0an =2n-1n=1.2,3--5E2n-1,\f(x) =is, + , cont. on [1,00)n=12x - 18x-af [1,00u=2x-lbdx = limdu=2 dxdx2x-1b-> 82X=du =/21X=u=1= lim 11 duX = bu=2b-1 1,"= lim00 1/1 du uX =bu=2b-1b-sbbblim In/2x-1)= lim1/2In I=21 2 b-oob-owillIID= 1/2 2 b-so (im[(2b-1)-lad)yinInb= 1/2 ?lim In(2b-1)bTinbbIn cont.InbTh-slim (2b-1)&= 1/2 Inbook= 1/2 In 2=8DHence,Ediv by IT2n-1n=18@ E6n sin 1/5 nn-asn=1sin touTOSlim = lim n sin 1/1 =Cim1/5ann-3001/2n -> &n-s D I'm un-n-sn&n-s D1/4= lim 4-70 sinu is=170nSinuHence, East ndiv by nth term testupositive termsIa265n(5n+1) & n=1,2,Vn ( nn +1)f(x)= I is + / v, cont onn=1X (Vx+1)[1,00]8bdx= limdxu=vx+1(5x +1)Vx +1)dxXb-> &[Xdu=/2S2 du= limu2du= dx 5x15b- &lim 2 In/u/I=2 2 lim In/Vx+1)=IIb->&b &IDInb= 2 lim[In (b+1) - Inz)bb&8 - - In28Hence, Statediv by IT Hence, Exting div by + /34, 8 n tan 1 n Apply nth term Test un=1tan1lim n tan 1/5=lim(im ann-s 00n->8boslimtan u0/0=uuse= limsecuseco = ==cosou to= to 1" 70Hence, (n tan 1/5 div by nth term testInnInn8 Use II to check Conu/DivIn8EInn'an Inn't postive termnfGI= Inx positive on [2,00]X cont. on [2,00n=2f(x) =/ X (2 1) - 2 Inx' -- 2- X 2InxXT ((x) =i2x1.xfie)= = 2-2heco2-1nx-0012=2Inx2f++++E1=1nxe2.7181e=e(e = X)D&bn t lnxi 1,(c)Inn'CtoInzf(x)= xEInn=+on [3,00]n2nn=2nE3J8nil= InzInn'=8+isn=3b88Stands = S Inx' Xdx= lim12haddXb. 8333u=lnxalbblim S 24 du = limdu= = dxb-so3bh-s32= lim(Inx)h-s 83 s2= lim(Unit), - (b)h-s & a - is= D8HenceE diu by ITn=38DHence,{Inn' = Inz tEluntdivnn=2n=3= Inz + 00= D-