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November 4th, 2014, 11:04 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, 12:49 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,566 Thanks: 2567 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, 12:55 PM  #3 
Math Team Joined: Jul 2011 From: Texas Posts: 2,805 Thanks: 1449 
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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comp, comparison, direct, limit, test 
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