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In my Calc II class, we're just starting convergence tests and all the examples are very convinient and they work perfectly (obviously, since they are examples), but my professor couldn't really answer my question.

Is there a time, or some examples, where convergence tests will contradict each other? Like the ratio test says the series will converge, but the limit comparison test says it diverges, etc.

And if there are examples, how would I know which one is correct?

Again, sorry if this is trivial...

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The more frequent case is that all but one test yields inconclusive results... –  J. M. Oct 31 '11 at 19:00
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No. Mathematics is consistent so, if you solve the same problem in two ways and don't make any errors, you will get the same answer. Of course, as JM says, it is quite common that you may try two methods and find that only one of them gives an answer, while the other is inconclusive. –  David Speyer Oct 31 '11 at 19:02
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@DavidSpeyer As far as we know, mathematics is consistent :) –  Thomas Andrews Oct 31 '11 at 19:07
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If math is inconsistent, see xkcd.com/816 for what to do next. –  David Speyer Oct 31 '11 at 19:15
    
If you get far enough with math to reach analysis, you'll be able to see why the answer is no. –  Lost Apr 25 at 22:50

1 Answer 1

up vote 4 down vote accepted

They should never contradict one another. The worst that could happen is that one test is inconclusive while another definitively shows that the limit does or does not exist.

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Well, the very worst that can happen is that all the tests we try are inconclusive... (-: –  Arturo Magidin Oct 31 '11 at 19:05
    
Ha, that's very true. –  AMPerrine Oct 31 '11 at 19:05

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