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

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SUMMARY

The expression sqrt(4n)/sqrt(4n-3)sqrt(4n^2-3n) diverges as it is asymptotically equivalent to 1/2n, which diverges like the harmonic series. A rigorous proof is required for real analysis, and the limit comparison test is recommended for this purpose. The limit comparison test simplifies the process by focusing on the limit of the ratio of the two sequences rather than requiring direct comparison for all terms.

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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.
 
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grossgermany said:

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?
 
Yes, comparison test would be great.
 
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 an >= bn for all n >= n0.

For the limit comparison test, you only need to look at lim an/bn.
 

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