 In this video, we provide the solution to question number nine for practice exam number four for math 1220 We're given a sequence a sub n is equal to 1 over n times the natural log of n We're also given a function f of x which is equal to 1 over x times the natural log of x Notice here that f of x is the continuous extension of the sequence a sub n there So it's the same formula. It's just one is a discrete sequence one is a continuous function and so we need to determine the convergence of the series where n equals to the sum n equals 2 to infinity of that sequence a n that is 1 over n times the natural log of n there We have to determine whether it's convergent or divergent and we're going to use the integral test here for which of course We'd be very interested in what happens to the integral of that function And so the first thing we want to do is evaluate the improper integral The integral from 2 to infinity of f of x dx here because the convergence of this thing will agree with the convergence of that So we're gonna we're gonna do that calculation so the integral from 2 to infinity of 1 over x times the natural log of x dx a U substitution would be very appropriate here if you take u to be the natural log of x Then du is equal to 1 over x dx and so applying that substitution You're gonna get a du on top and then a u in the denominator I also want to change the bounds Because when x is 2 you're gonna get the natural log of 2 that's not so bad when x goes towards infinity You're gonna get u is also approaching infinity like so and so then continuing on here You're gonna take the antiderivative of 1 over u which is the natural log of the absolute value of u and you're gonna evaluate this As you go from the natural log of 2 to infinity here if I plug in the natural log of 2, there's no problem there You're just gonna get the log log 2 The problem is when you plug in infinity here, of course, you're gonna take the natural log of infinity For which the natural log of infinity is itself infinite and so this thing is gonna become Infinity minus a number But that number is peanuts compared to infinity this thing is gonna be infinite so we just calculated that the integral is actually infinite so put your final answer here on the line and By the integral test the convergence of this integral is the same as the convergence of this series The improper integral is divergent because it became infinity so the series is likewise divergent by the integral test