My Math Forum How do I know when to use the Limit Comparison Test over Direct Comp. Test?

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 November 4th, 2014, 10:04 AM #1 Member   Joined: Oct 2013 Posts: 38 Thanks: 2 How do I know when to use the Limit Comparison Test over Direct Comp. Test? If I have An = 2 / root(n^2 - 3), Bn = 1/n I assume that I would probably use the LCT, instead of the direct test, but why couldn't I just use the direct test? is the 2 in the numerator that much of a factor in the long run? More so than the -3, or -4, or whatever it was that would make the number bigger? The distinction of when to use one over the other is confusing me right now.
 November 4th, 2014, 11:49 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra The 2 makes no difference to me. $$a_n = \frac{2}{\sqrt{n^2 - 3}} \gt \frac2n \gt \frac1n = b_n \qquad n \gt 1$$
 November 4th, 2014, 11:55 AM #3 Math Team     Joined: Jul 2011 From: Texas Posts: 3,002 Thanks: 1588 in the case where you might have $\displaystyle a_n = \frac{2}{\sqrt{n^2+3}}$ , you cannot do a direct comparison with $\displaystyle \frac{2}{n}$ since $\displaystyle \frac{2}{\sqrt{n^2+3}} < \frac{2}{n}$ ... in which case, use the limit comparison test.

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