Question

please show all steps

Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. 2 Σ k(Ink)? k= 3 Select the correct choice below and, if necessary, fill in the answer box to complete the choice. O A. 2. The series converges. The value of the integralſ dx is x(In x)2 (Type an exact answer.) B. 2 The series diverges. The value of the integralſ -dx is x( In x)2 (Type an exact answer.) O C. The Integral Test does not apply.

Answer 1

The given series is convergence by using integral test.
Given series is 2 I f(K) K: 3K (ink) К?3 where tlk) = x (inck,jo, k23 det tex)= 2 a(in (XJ) " 7,823 elearly fx) xo for an xt [3,00). Let x, x26 [3,00) such that xi Xa(In (*2))"> i(In (m1))? 212 Then -> > az[m(*2))" sum (84)) Therefore f(x) > free) for all 2,026 [3,0) where y cu2 Therefore the function f(x) monotone deen leeresing on the interval [3,0). Now using integral test substitute dn 3 K (line) noz keln(x))" u = ln (2) dua t de alim jheng In (n) 2 du ulin (3/m(n) u² m(3 n 2 $ 2 x 3 n

Joncos In(n) tur du ท10 = lim a 'in (3) in(n) lim (2) in (3) ü 2du ni lim hyoo 2 = lim (2) tim (2) [w-moto (2) [-them theos] [0+ tnc39] of In(ou)? (finite value) Therefore the improper integral Ja convergent. in (3) dn= 2 In(3) dn is w 13 x ln cuja de 123 *((K))? is convergent Therefore the given series by using integral test. Answer: ies. The value of the The series converges. integral Jour he is a la ea gazofre