Use the Limit Comparison Test to determine the series' convergence?

[EnlightenedOne]

Homework Statement

Use the Limit Comparison Test to determine if the series converges or diverges:
Ʃ (4/(7+4n(ln^2(n))) from n=1 to ∞.

(The denominator, for clarity, in words is: seven plus 4n times the natural log squared of n.)

Homework Equations

Limit Comparison Test:
Let Σa(n) be the original series, and Σb(n) be the comparison series.
1. If the lim(n->∞) a(n)/b(n) is a positive number, but not ∞, then a(n)'s convergence/divergence is the same as b(n)'s.
2. If the lim(n->∞) a(n)/b(n) = 0, then if b(n) is convergent, so is a(n), but if b(n) is divergent, the result is inconclusive.
3. If the lim(n->∞) a(n)/b(n) = ∞, then if b(n) is divergent, so is a(n), but if b(n) is convergent, the result is inconclusive.

The Attempt at a Solution

I need to use a p-series to determine if this series diverges or converges, but every p-series I have tried gives the result that makes it inconclusive.

Here is the issue:
When I make b(n) = 1/n (p-series that diverges), I get a limit of 0, which is inconclusive (via case 2).
When I make b(n) = 1/(n^2) (p-series that converges), I get a limit of ∞, which is inconclusive (via case 3).

No matter what p-series I try, I get an inconclusive case...

Can anyone find a comparison p-series that (using the Limit Comparison Test) works?? My instructor said a p-series will work, but everything I try does not.

Please show all steps. Thank you!

 

[LCKurtz]

I don't know about the p series thing, but you might try a limit comparison test with $b_n=\frac 1 {n\ln^2 n}$ which series you can try the integral test on. You may have to start at $n=2$ for the second series and integral test.

 

[EnlightenedOne]

I don't know about the p series thing, but you might try a limit comparison test with $b_n=\frac 1 {n\ln^2 n}$ which series you can try the integral test on. You may have to start at $n=2$ for the second series and integral test.

I know what you are saying, but my instructor is expecting a p-series, and I have to use the Limit Comparison Test. I really don't know why my instructor says that there is one, but that has to mean there is, unless there's a mistake.

What do I do if no others reply? Can I re-post this thread?

 

[EnlightenedOne]

My instructor must be mistaken, because everyone that replies to my forum threads on this question says to try what you said. I'll just do that and explain it to my instructor next class.

Thank you