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Math Help - alternate series question

  1. #1
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    alternate series question

    Converges or diverges?

    SUM n_infinity (-1)^n(2^n*n!)/(5*8*11*...*(3n +2)

    i believe this is an alternating series.

    the 5*8*11*.. confuses me.

    thankyou for any help.
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  2. #2
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    Hello, rcmango!

    Converges or diverges?

    SUM n_infinity (-1)^n(2^n*n!)/(5*8*11*...*(3n +2)

    I would use the Ratio Test


    a
    n+1 . . . . .2^n+1(n+1)! . . .5811(3n+2) . . . .2(n + 1) . . . . 2n + 2
    ------ . = . ------------------ ------------------ . = . ---------- . = . ---------
    . a
    n . . . . .5811(3n+5) . . . 2^nn! . . . . . . . . 3n + 5 . . . . . 3n + 5


    . . . . . . . . . . . . . . . . . . . . . 2 + 2/n
    Divide top and bottom by n: . ---------
    . . . . . . . . . . . . . . . . . . . . . 3 + 5/n


    Now take the limit as n → ∞

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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Soroban View Post
    Hello, rcmango!


    I would use the Ratio Test


    an+1 . . . . .2^n+1(n+1)! . . .5811(3n+2) . . . .2(n + 1) . . . . 2n + 2
    ------ . = . ------------------ ------------------ . = . ---------- . = . ---------
    . an . . . . .5811(3n+5) . . . 2^nn! . . . . . . . . 3n + 5 . . . . . 3n + 5


    . . . . . . . . . . . . . . . . . . . . . 2 + 2/n
    Divide top and bottom by n: . ---------
    . . . . . . . . . . . . . . . . . . . . . 3 + 5/n


    Now take the limit as n → ∞
    Now this is overkill, we have proven that the series is absolutely convergent.

    We could have proven that it converges (without proving that it is
    absolutely convergent) by using the Alternating Series test.

    This requires that we show that the absolute value of the terms is
    eventually decreasing, and the limit of the terms is 0.

    RonL
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by Soroban View Post
    Hello, rcmango!


    I would use the Ratio Test


    an+1 . . . . .2^n+1(n+1)! . . .5811(3n+2) . . . .2(n + 1) . . . . 2n + 2
    ------ . = . ------------------ ------------------ . = . ---------- . = . ---------
    . an . . . . .5811(3n+5) . . . 2^nn! . . . . . . . . 3n + 5 . . . . . 3n + 5
    This seems to have lost the alternating signs of the original series, so is
    actually

    |a_{n+1}/a_n|

    and so you have proven absolute convergence

    RonL
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