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Thread: Bounded sequence

  1. #1
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    Bounded sequence

    So I have this problem I don't know how to solve:

    "Show that the following sequences are bounded:"

    $\displaystyle U_n = \frac{2n}{3n+1}$

    $\displaystyle V_n = sin(\frac{\pi \times n}{2}) + cos(\pi \times n)$

    Help would be appreciated!
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  2. #2
    MHF Contributor red_dog's Avatar
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    Obviously, $\displaystyle U_n\geq 0$.

    We prove that $\displaystyle U_n<\frac{2}{3}$.

    $\displaystyle \frac{2n}{3n+1}<\frac{2}{3}\Leftrightarrow 6n<6n+2\Leftrightarrow 0<2$

    Then $\displaystyle 0\leq U_n<\frac{2}{3}, \ \forall n\in\mathbb{N}$

    Sin and Cos are bounded by -1 and 1. Then their sum is bounded by -2 and 2.

    Or,

    $\displaystyle V_n=\left\{\begin{array}{llll}1, & n=4k\\0, & n=4k+1\\1, & n=4k+2\\-2, & n=4k+3\end{array}\right.$

    Then $\displaystyle -2\leq V_n\leq 1, \ \forall n\in\mathbb{N}$
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