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Math Help - Series/Convegence tests! need help on review problems for test!

  1. #1
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    Series/Convegence tests! need help on review problems for test!

    1. A certain ball has the property that each time it falls from a height h onto a hard, level surface, it rebounds to a height rh, where 0 < r < 1. Suppose that the ball is dropped from an initial height of H meters.
    a) Assuming the ball bounces indefinitely, find the total distance that it travels (Use the fact that the ball falls 1/2gt^2 meters in t seconds)
    b)calculate the total time that the ball travels

    2. A series is Sum an is defined by the equations:

    a1 = 1 an+1 = [(2+cosn)/(sqrt(n)]*an

    Determine whether Sum an converges or diverges.

    could someone show me how to do these? going NUTS.
    thank u
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  2. #2
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    Hello, xfyz!

    I'll get you started on #1 . . .


    1. A certain ball has the property that each time it falls from a height h
    onto a hard, level surface, it rebounds to a height rh, where 0 < r < 1.
    Suppose that the ball is dropped from an initial height of H meters.

    a) Assuming the ball bounces indefinitely, find the total distance that it travels.
    Let's baby-talk our way through the first few bounces . . .


    First, it falls H meters.

    Then it bounces up Hr
    m . . . and falls Hr m.

    Then it bounces up (Hr)r = Hr^2 . . . and falls Hr^2

    Then it bounces up (Hr^2)r = Hr^3 . . . and falls Hr^3

    . . and so on . . .


    The total distance is: . D \;=\;H + 2Hr + 2Hr^2 + 2Hr^3 + \cdots

    . . \text{and we have: }\;D \;=\;H + 2Hr\underbrace{\left(1 + r + r^2 + r^3 + \cdots\right)}_{\text{geometric series}}

    The series in paretheses has first term 1 and common ratio r.
    . . Its sum is: . \frac{1}{1-r}

    So we have: . D \;=\;H + 2Hr\left(\frac{1}{1-r}\right)\quad\Rightarrow\quad\boxed{D \;=\;H\left(\frac{1+r}{1-r}\right)}

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  3. #3
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    Quote Originally Posted by xfyz View Post
    2. A series is Sum an is defined by the equations:

    a1 = 1 an+1 = [(2+cosn)/(sqrt(n)]*an

    Determine whether Sum an converges or diverges.

    could someone show me how to do these? going NUTS.
    thank u
    I think is converges.

    First note that,
    a_{n+1} = \left( \frac{2+\cos n}{\sqrt{n}} \right) \left( \frac{2+\cos (n-1)}{\sqrt{n-1}} \right) ... \left( \frac{2+\cos 1}{\sqrt{1}}\right).
    Thus,
    |a_{n+1}| = \left| \frac{2+\cos n}{\sqrt{n}}\right| \cdot ... \cdot \left| \frac{2+\cos 1}{\sqrt{1}} \right| \leq \frac{3}{\sqrt{n}}\cdot ...\cdot \frac{3}{\sqrt{1}} = \frac{3^n}{\sqrt{n!}}
    Use the root test,
    \left| \frac{3^n}{\sqrt{n!}}\right|^{1/n} = \frac{3}{\sqrt{(n!)^{1/n}}} \to 0 because (n!)^{1/n} \to \infty.
    Thus, by the comparison test this series converges.
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