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Math Help - determine whether the sequence converrger or diverges

  1. #1
    Junior Member
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    determine whether the sequence converrger or diverges

    Here is the problem:
    an = n/(1+sqrt(n))
    I divided top and bottom by n
    then had 1 in the numerator
    and 1/n + 1/n^(1/2) in the denominator
    could I think about this like it is 1/0 ?
    because there is still an n in the denominator
    so the sequences diverges, is this correct?
    The problems with n^2 or n seem so easy but this one where it is sqrt(n) or N^(1/2) gives me trouble.

    Thank you,
    Keith
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  2. #2
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    Hello, Keith!

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

    Divide top and bottom by \sqrt{n} .**

    We have: . \frac{\dfrac{n}{\sqrt{n}}\quad}<br />
{\dfrac{1}{\sqrt{n}} + \dfrac{\sqrt{n}}{\sqrt{n}}} \;=\;\frac{\sqrt{n}}{\dfrac{1}{\sqrt{n}} + 1}


    Then: . \lim_{n\to\infty}\left(\frac{\sqrt{n}}{\frac{1}{\s  qrt{n}} + 1}\right) \;=\;\frac{\infty}{0 + 1} \;=\;\infty . . . diverges


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    **

    Divide through by the highest power of the variable in the denominator.

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