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Math Help - Absolute Converging Summation

  1. #1
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    Absolute Converging Summation

    Hi, I'm in precalc and were reviewing trig right now and i'm bored out of my mind. When I was doing a question on my homework I noticed something interesting in one of the questions. I realize that I can prove the answer with trig in about 5 seconds flat but I would like to understand how to use an infinite series sum to describe the relationship first, and then figure out how to find the limit of the series to get my answer as opposed to doing a little trig. Here is a picture of the problem and my thoughts(sorry for the poor quality, all the school has is ms paint :/):


    ImageShack - Hosting :: problemdj9.png

    ImageShack - Hosting :: limitfm0.png

    So height z is the sum of the infinite series y+y_1+...+y_n
    First, how would I go about writing the summation to describe the y_n's?
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by gtozoom View Post
    Hi, I'm in precalc and were reviewing trig right now and i'm bored out of my mind. When I was doing a question on my homework I noticed something interesting in one of the questions. I realize that I can prove the answer with trig in about 5 seconds flat but I would like to understand how to use an infinite series sum to describe the relationship first, and then figure out how to find the limit of the series to get my answer as opposed to doing a little trig. Here is a picture of the problem and my thoughts(sorry for the poor quality, all the school has is ms paint :/):


    ImageShack - Hosting :: problemdj9.png

    ImageShack - Hosting :: limitfm0.png

    So height z is the sum of the infinite series y+y_1+...+y_n
    First, how would I go about writing the summation to describe the y_n's?
    Your second link does not work. Just fill in the follwoing

    \sum_{n=a}^{\infty}y_n

    a=?

    y_n=?
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  3. #3
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    Ok, I got the summation and I did it out to 10 places and it was pretty damn close so I'm assuming it's right. How to I find a close-formed solution to it though?
    Z=y_0+\sum_{n=1}^\infty \frac{y_{n-1}}{1.92381}

    i found it by noticing that the triangles are all similar and that the y_n's are decreasing proportionately by 1/1.92381*the last number in the sequence
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by gtozoom View Post
    Ok, I got the summation and I did it out to 10 places and it was pretty damn close so I'm assuming it's right. How to I find a close-formed solution to it though?
    Z=y_0+\sum_{n=1}^\infty \frac{y_{n-1}}{1.92381}

    i found it by noticing that the triangles are all similar and that the y_n's are decreasing proportionately by 1/1.92381*the last number in the sequence
    If the ratio is 1.92... then you would have

    S=\sum_{n=0}^{\infty}y_0\left(1.92...\right)^n

    Which diverges to infinity.
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