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April 7th, 2013, 01:09 PM   #1
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Comparison, Limit Comparison, and Splitting Sums.

I'm not really sure when each of these should be done. In fact, I don't really understand the reason that we use the limit comparison test.


So here I can simply say that P=2>1, so the original converges.


Here, I would say that P=3>1, implying the original converges. But my solutions tell me that here I should use a limit comparison with 1/N^3, where the limit ---->N is = 1. So why did I do this? What did I prove by finding the limit that I didn't prove by just comparing directly? What is different in this that I had to use a limit comparison instead of direct? Just the added N^2? I don't understand why.

And I also see that when I have some questions that have similar sums to the one directly above me, they just split the sum and evaluate the p series of both sums to find the divergence; is this an alternative to a limit comparison test as well?

Thank you, guys.
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April 18th, 2013, 07:05 AM   #2
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Re: Comparison, Limit Comparison, and Splitting Sums.

[color=#000000]For the first one since , , so the series converges.

For the second one here is another way, you can compute the result by doing the following

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