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    So I'm trying to work out the following questions:
    If n(an) tends to 0 as n tends to infinity, must sigma(an) converge?
    If n^2(an) tends to 0 as n tends to infinity, must sigma(an) converge?
    Give proofs or counter-examples.

    All I can do myself is say that mod(nan) less than epsilon for n greater than or equal to N, and deduce that mod(an) tends to 0. While this obviously indicates that sigma(an) could converge, it definitely doesn't prove that it does converge.

    Any help please???
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    I recommend solving the second one first. If n^2 an goes to 0 then what is an eventually less than?
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    I'll take this thread as an opportunity to point out how you can use "what would be a sensible pair of questions?" to decide what the likely answers are before you do any significant mathematics:

    If n^2 a_n \to 0 then certainly n a_n \to 0. So, if the first statement was true, you wouldn't have to do any work to show the 2nd statement to be true. It seems unlikely an examiner would let this happen.

    Conversely, if the 2nd statement was false, then a counterexample for the 2nd statement would be a counterexample for the 1st statement. Again, it seems unlikely an examiner would let this happen.

    So by "exam logic", you'd expect the 1st statement to be false and the 2nd statement to be true.
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    (Original post by DFranklin)
    I'll take this thread as an opportunity to point out how you can use "what would be a sensible pair of questions?" to decide what the likely answers are before you do any significant mathematics:

    If n^2 a_n \to 0 then certainly n a_n \to 0. So, if the first statement was true, you wouldn't have to do any work to show the 2nd statement to be true. It seems unlikely an examiner would let this happen.

    Conversely, if the 2nd statement was false, then a counterexample for the 2nd statement would be a counterexample for the 1st statement. Again, it seems unlikely an examiner would let this happen.

    So by "exam logic", you'd expect the 1st statement to be false and the 2nd statement to be true.
    Okay, thanks. That's what I thought, but I've been finding it really hard to come up with a counterexample for the first one or a proof for the second. Does anyone have any tips on that?
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    (Original post by 4321)
    Okay, thanks. That's what I thought, but I've been finding it really hard to come up with a counterexample for the first one or a proof for the second. Does anyone have any tips on that?
    Well I gave you advice for the second one already. Once you've got that you can see what kind of counterexample the first one will require by looking at where the proof breaks down.
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    I've got it! Thank you so much!!
 
 
 
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