Sqrt(4n)/sqrt(4n-3)sqrt(4n^2-3n) diverges?

[grossgermany]

Homework Statement

How to show that sqrt(4n)/sqrt(4n-3)sqrt(4n^2-3n) diverges?

Homework Equations

The Attempt at a Solution

The above expression is asymptotically equivalent to 1/2n which diverges as the harmonic series diverges.

However, a rigorous proof is required for the real analysis class.

 

[Mark44]

Homework Statement

How to show that sqrt(4n)/sqrt(4n-3)sqrt(4n^2-3n) diverges?

Homework Equations

The Attempt at a Solution

The above expression is asymptotically equivalent to 1/2n which diverges as the harmonic series diverges.

However, a rigorous proof is required for the real analysis class.

How rigorous? Would the limit comparison test be rigorous enough?

 

[grossgermany]

Yes, comparison test would be great.

 

[Mark44]

I would use the limit comparison test, not the comparison test. For the comparison test, and some series $\sum b_n$ that is known to diverge, you would have to show that a_n >= b_n for all n >= n₀.

For the limit comparison test, you only need to look at lim a_n/b_n.