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
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 :)