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November 4th, 2014, 10:04 AM   #1
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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.
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November 4th, 2014, 11:49 AM   #2
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The 2 makes no difference to me.

$$a_n = \frac{2}{\sqrt{n^2 - 3}} \gt \frac2n \gt \frac1n = b_n \qquad n \gt 1$$
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November 4th, 2014, 11:55 AM   #3
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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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