Lecture Note
University
California State UniversityCourse
MATH 150B | Calculus IIPages
8
Academic year
2023
alifah chan
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0
Integral Test (Conv/Div, IT ? 8 an Consider an / where n=1 n=K v.an positive terms and in f(n) is cont. , +1 on [K, o) in 8 Then Ean 8 an and Sf(x)dx both Conv. or n=K K both div Exp Does the following Series Conv. / Div ? & . an= 1/2 positive term I n=1,2,3 1 E . f(x)= 1/X is +, cont., on [1,00) n n=1 8 8 dx = \ S = 1=1 Hence, E 1/2 con to / n2 = 2J-1 n=1 I by IT D IT 2 E Con by n3 n=1 conv ~ r 0 1 1
conv & Exp+ dx X XP = P-1 if p>1 I 8 if P I I Div com Exp dr = if p
1 positive terms & 00 . an n'+1 4 E n2+1 I n=1 f(x)= is +, cont, on [1,00) x + +1 8 b b dx = lim dx = lim tan x x' + 1 2 b-> as x2 +1 b-so I = lim tan b - tan'i] 17/47 tan'b b-ow = H2 - IF / b I'll - 2 II = 4 Hence, Conv by IT n°+1 n=1 is postive for all 0 an = 2n-1 n=1.2,3-- 5 E 2n-1, \ f(x) = is , + , cont. on [1,00) n=1 2x - 1 8 x-af [1,00 u=2x-l b dx = lim du=2 dx dx 2x-1 b-> 8 2X = du = / 2 1 X = u=1 = lim 11 du X = b u=2b-1
1, " = lim 00 1/1 du u X = b u=2b-1 b-s b bb lim In/2x-1) = lim 1/2 In I = 2 1 2 b-oo b-ow ill I ID = 1/2 2 b-so (im [ (2b-1) - lad) y in Inb = 1/2 ? lim In(2b-1) bT inb b In cont. Inb T h-s lim (2b-1) & = 1/2 In book = 1/2 In 2 = 8 D Hence, E div by IT 2n-1 n=1 8 @ E 6 n sin 1/5 n n- as n=1 sin to u TOS lim = lim n sin 1/1 = Cim 1/5 an n-300 1/2 n -> & n-s D
I'm un - n-s n & n-s D 1/4 = lim 4-70 sinu is =1 70 n Sinu Hence, East n div by nth term test u positive terms I a 26 5n(5n+1) & n=1,2, Vn ( nn +1) f(x)= I is + / v, cont on n=1 X (Vx+1) [1,00] 8 b dx = lim dx u=vx+1 (5x +1) Vx +1) dx X b-> & [X du= / 2 S 2 du = lim u 2du= dx 5x 15 b- & lim 2 In/u/ I =2 2 lim In/Vx+1) = II b-> & b & ID Inb = 2 lim [ In (b+1) - Inz) b b & 8 - - In2 8 Hence, State div by IT
Hence, Exting div by + / 34, 8 n tan 1 n Apply nth term Test u n=1 tan 1 lim n tan 1/5 = lim (im an n-s 00 n-> 8 bos lim tan u 0/0 = u use = lim secu seco = = = coso u to = to 1" 70 Hence, (n tan 1/5 div by nth term test Inn Inn 8 Use II to check Conu/Div I n 8 E Inn' an Inn't postive term n fGI= Inx positive on [2,00] X cont. on [2,00 n=2 f(x) = / X (2 1) - 2 Inx' -- 2- X 2 Inx X T
((x) = i 2 x 1.x fie)= = 2-2heco 2-1nx-00 1 2 =2 Inx 2 f ++++ E 1=1nx e 2.718 1 e=e (e = X) D & bn t lnxi 1,(c) Inn' C to Inz f(x)= x E Inn = + on [3,00] n 2 n n=2 nE3 J 8 nil = Inz Inn' = 8 + is n=3 b 8 8 Stands = S Inx' X dx = lim 12hadd X b. 8 3 3 3 u=lnx al b b lim S 24 du = lim du= = dx b-so 3 b h-s 3 2 = lim (Inx) h-s 8 3
s 2 = lim (Unit), - (b) h-s & a - is = D 8 Hence E diu by IT n=3 8 D Hence, { Inn' = Inz t E lunt div n n=2 n=3 = Inz + 00 = D -
Integral Test
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