00:02
In this question, we are asked to determine the convergence of the following series.
00:07
2 over k, ln k squared, where k runs from 2 to infinity.
00:23
To determine conversions with the series, we are going to use an integral test.
00:31
So let f of x be equal to 2 over x l n x squared.
00:40
Remember to apply the integral test we want our function f of x to be continuous and decreasing.
00:50
Clearly it's continuous for for x greater than or equal than two and it's decreasing for x greater than 2.
01:15
That's true because whenever you start plugging in bigger numbers the fraction gets smaller and smaller.
01:24
For example, when you plug in x equals 10, we will get 2 over 10 a 10 squared, and that's going to be smaller than 2 over 1 million a len 1 million squared.
01:36
So this function is decreasing, this continues, meaning we can apply integral test.
01:53
The integral test says that if the integral, if the improper integral of f -of -x -d -x converges, then a our series converges.
02:11
If the integral diverges, then our series diverges.
02:19
Thus we have to calculate this integral 2 over x, when x squared x.
02:32
Since this integral is improper, we need to use limits...