# How do you do the limit comparison test for this problem #sqrt ( (n+1)/ (n^2+2))# as n goes to infinity?

##### 2 Answers

Diverges when compared to

#### Explanation:

We need to come up with a new sequence

So, let's create

Now, we know

The Limit Comparison tells us if we know the convergence or divergence of

Knowing

Then, both series diverge.

The series diverges.

See work below:

#### Explanation:

The idea of the limit comparison test is that you essentially compare your unknown function to a function whose convergence you know (through another method: typically p-test). Here's how you do this:

Let's say the series you want to analyze is

If this limit

If this limit

If this limit

So now, we figure out what series it would be ideal to compare this to. I'm going to chose **By the p-test, we know that this series diverges.** Bearing this in mind, let's take the limit:

Now, we just evaluate this limit using the same steps we learned in Calc 1. We just divide every term by the highest power:

..and now take the limit as

This limit is neither 0 nor infinity, but it's a finite value (

Hope that helped :)