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Math Help - Nature of the sequence

  1. #1
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    Nature of the sequence

    I have already posted this question but I accidentally asked for the nature of the 'series', where as I wanted someone to help me with,
    how I can tell if the following sequences {an} diverge or converge? And how to justify it?
    Nature of the sequence-question-1-assignment-2.png
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  2. #2
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    Re: Nature of the sequence

    Hey calmo11.

    The first thing you should do is list the type of series (alternating, non-alternating in sign) and list the different kinds of tests for that particular series. Can you do this to start off with?
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  3. #3
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    Re: Nature of the sequence

    Hello, calmo11!

    How I can tell if the following sequences diverge or converge? And how to justify it?
    Take the limit of a_n as n\to\infty.
    If the limit is finite, the sequence converges.



    (a)\;a_n \:=\:\frac{(\text{-}1)^n}{n}

    We see that: . \displaystyle \lim_{n\to\infty}\frac{(\text{-}1)^n}{n} \;=\;0

    The sequence converges.




    (b)\;a_n \:=\:\frac{(\text{-}1)^n + n}{(\text{-}1)^n - n}

    Divide numerator and denominator by n\!:\;\;\frac{\dfrac{(\text{-}1)^n}{n} + 1}{\dfrac{(\text{-}1)^n}{n} - 1}

    Then: . \lim_{n\to\infty}\frac{\dfrac{(\text{-}1)^n}{n} + 1}{\dfrac{(\text{-}1)^n}{n} - 1} \;=\;\frac{0+1}{0-1} \;=\;-1

    The sequence converges.




    (c)\;a_n \:=\:\sqrt{n+1} - \sqrt{n}

    Multiply by \tfrac{\sqrt{n+1} + \sqrt{n}}{\sqrt{n+1}+\sqrt{n}}\!:\;\;

    . . \frac{\sqrt{n+1} - \sqrt{n}}{1}\cdot\frac{\sqrt{n+1} + \sqrt{n}}{\sqrt{n+1} + \sqrt{n}} \;=\; \frac{(n+1) - n}{\sqrt{n+1} + \sqrt{n}} \;=\;\frac{1}{\sqrt{n+1} + \sqrt{n}}

    Then: . \lim_{n\to\infty}\frac{1}{\sqrt{n+1}+\sqrt{n}} \;=\;\frac{1}{\infty} \:=\:0

    The sequence converges.
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