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Math Help - Geometric series

  1. #1
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    Geometric series

    Find the sum to n terms of this series:

    1 -y +y^2 -y^3 ...

    I have figured that a=(-1)^n+1.
    r=-y.
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  2. #2
    Super Member Quacky's Avatar
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    You are correct in saying that r=-y, but you're overcomplicating the other part: a is just the first term in this series. In this case, a = 1. Now you just need to substitute the values into the formula.
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  3. #3
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    That works but it isn't the answer in the book.

    It is:

    1+((-1)^n+1) * y^n
    divided by 1+y.
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  4. #4
    Super Member Quacky's Avatar
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    Hmm...have you posted the question correctly? I can't see any error in my logic.
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  5. #5
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    You and the book are correct. The answer in the book confused me. They must have decided to take the answer a step further.
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  6. #6
    MHF Contributor harish21's Avatar
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    You are considering r=-y. And the first term of your series is :a= 1.

    so the sum of the geometric series would be given by:  S = \frac{a(1-{r^n})}{1-r}

    try plugging in the values now
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  7. #7
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    I've already done that after the first post by Quacky.
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  8. #8
    Super Member Quacky's Avatar
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    I think the textbook is either misprinted or has found some other values which also work, but I don't believe my calculation is incorrect.
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  9. #9
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    Quote Originally Posted by Stuck Man View Post
    Find the sum to n terms of this series:

    1 -y +y^2 -y^3 ...

    I have figured that a=(-1)^n+1.
    r=-y.
    a=1

    r=-y

    S_n=\displaystyle\frac{a\left(1-r^n\right)}{1-r}=\frac{a\left(r^n-1\right)}{r-1}

    =\displaystyle\frac{1-r^n}{1-r}=\frac{r^n-1}{r-1}

    =\displaystyle\frac{1-(-y)^n}{1-(-y)}=\frac{(-y)^n-1}{-y-1}

    =\displaystyle\frac{1-(-1)^ny^n}{1+y}=\frac{(-1)^ny^n-1}{-(y+1)}

    =\displaystyle\frac{1-(-1)^ny^n}{1+y}=\frac{1+(-1)(-1)^ny^n}{1+y}=\frac{1+(-1)^{n+1}y^n}{1+y}
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  10. #10
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    Thats more work than necessary. You don't need to have loads of steps to produce the coefficient of -1 or 1. Geometric series-core15c7cans.jpg
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  11. #11
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    Quote Originally Posted by Stuck Man View Post
    Thats more work than necessary. You don't need to have loads of steps to produce the coefficient of -1 or 1. Click image for larger version. 

Name:	Core15c7cAns.jpg 
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    The steps are only for clarity.
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