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Math Help - Nature of the series

  1. #1
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    Nature of the series

    Could someone tell me how I can tell if the following sequences {an} diverge or converge? And how to justify it?Nature of the series-question-1-assignment-2.png
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  2. #2
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    Re: Nature of the series

    Calmo

    liman = 0 , lim b = -1 and lim c = 0

    very easy to ustify.....
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  3. #3
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    Re: Nature of the series

    There is some confusion, your title reads series but your post reads sequences... i'm going to assume you want series behavior.

    a)Convergent. It is an alternating series,  a_n \to 0 as  n \to \infty

    b) Divergent  a_n \to -1 as  n \to \infty , its like adding -1 infinite times

    c) divergent but a bit harder to show, multiply numerator and denominator by the conjugate. consider the following...

     a_n \ = \ \frac{(\sqrt{n + 1} - \sqrt{n}) (\sqrt{n + 1} + \sqrt{n}) }{ \sqrt{n + 1} + \sqrt{n}}

     a_n \ = \ \frac{1}{\sqrt{n + 1} + \sqrt{n}}

    Now compare this with  b_n \ = \ \frac{1}{3 \sqrt{n}} which diverges because the exponent on n is less than 1

     a_n > b_n for n > 0 so a_n diverges because every nth term of a_n is LARGER than every corresponding nth term of b_n

    Last edited by agentmulder; April 14th 2013 at 08:26 PM.
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