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Thread: limits of sequences: u_n = [(2n^3 - 4n^2 + 5)/(10n^3 + 100)]*2^{-n), etc

  1. #1

    limits of sequences: u_n = [(2n^3 - 4n^2 + 5)/(10n^3 + 100)]*2^{-n), etc

    hello

    I really troubled with the following questions:



    [tex]\mbox{1. }\, u_n\, =\, \left(\dfrac{2n^3\, -\, 4n^2\, +\, 5}{10n^3\, +\, 100}\right)\, \cdot\, 2^{-n}[/tex]

    [tex]\mbox{2. }\, w_n\, =\, \left(\dfrac{3n\, +\, 2}{4n^2\, +\, 8n\, +\, 5}\right)\, \cdot\, \left(\dfrac{(1\, -\, n)^3}{(14\, -\, 5n)^2}\right)[/tex]

    [tex]\mbox{3. }\, e_n\, =\, \dfrac{2^n\, +\, 3^{n-1}\, +\, 5^{2n+2}}{4^{n-7}\, +\, 5^{2n}}[/tex]

    [tex]\mbox{4. }\, b_n\, =\, \dfrac{\left(\frac{2}{3}\right)^n}{\left(\frac{1}{ 2}\right)^n\, +\, \left(\frac{9}{10}\right)^n}[/tex]

    [tex]\mbox{5. }\, S_n\, =\, 4\, \sqrt{\strut n\, +\, 3\,}\, -\, \sqrt{\strut n\, -\, 1\,}\, -\, 5\, \sqrt{\strut n\, +\, 7\,}\, +\, 2\, \sqrt{\strut n\, -\, 3\,}[/tex]



    I need to find the limit of each sequence.

    My main concrn is thos 3 last sequence.
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    Last edited by stapel; 12-19-2017 at 06:09 PM. Reason: Typing out the text in the graphic; creating useful subject line.

  2. #2
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    Quote Originally Posted by someone111888 View Post
    hello

    I really troubled with the following questions:



    [tex]\mbox{1. }\, u_n\, =\, \left(\dfrac{2n^3\, -\, 4n^2\, +\, 5}{10n^3\, +\, 100}\right)\, \cdot\, 2^{-n}[/tex]

    [tex]\mbox{2. }\, w_n\, =\, \left(\dfrac{3n\, +\, 2}{4n^2\, +\, 8n\, +\, 5}\right)\, \cdot\, \left(\dfrac{(1\, -\, n)^3}{(14\, -\, 5n)^2}\right)[/tex]

    [tex]\mbox{3. }\, e_n\, =\, \dfrac{2^n\, +\, 3^{n-1}\, +\, 5^{2n+2}}{4^{n-7}\, +\, 5^{2n}}[/tex]

    [tex]\mbox{4. }\, b_n\, =\, \dfrac{\left(\frac{2}{3}\right)^n}{\left(\frac{1}{ 2}\right)^n\, +\, \left(\frac{9}{10}\right)^n}[/tex]

    [tex]\mbox{5. }\, S_n\, =\, 4\, \sqrt{\strut n\, +\, 3\,}\, -\, \sqrt{\strut n\, -\, 1\,}\, -\, 5\, \sqrt{\strut n\, +\, 7\,}\, +\, 2\, \sqrt{\strut n\, -\, 3\,}[/tex]



    I need to find the limit of each sequence.

    My main concrn is thos 3 last sequence.
    Unfortunately, you have not shown us any work of your own. You must have something or it is entirely unfair to have given you these problems on an assignment.

    Think about dominance.

    For example:
    1) Who cares what n^2 does if there is an n^3 in there?
    2) Does it really matter what (1/8)^n does if there us a (8/9)^n right next to it?
    3) If part of an expression converges to a finite value other than zero, and its neighbor converges to zero, what can we say of the entire expression?

    No hard and fast rules, here. Waiting for you to show us where you are.

    Try to remember that Convergence has nothing to do with a couple of terms (read: Finite Number of Terms) at the beginning. We are examining tail behavior.
    Last edited by stapel; 12-19-2017 at 06:10 PM. Reason: Copying typed-out graphical content into reply.
    "Unique Answers Don't Care How You Find Them." - Many may have said it, but I hear it most from me.

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