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Math Help - Finding Particular P-series

  1. #1
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    Finding Particular P-series

    Hello,

    So, I have this problem that is wanting me to find two particular p-series that satisfy some particular conditions. I don't really know how to approach this problem and I'm not sure I even understand what is being asked.

    The problem:

    "Find two p-series: \sum_{k=1}^{\infty}c_k and \sum_{k=1}^{\infty}d_k such that \sum_{k=1}^{\infty}\frac{8(-1)^kk-7(k)^\frac{1}{2}}{k^\frac{3}{2}} = \sum_{k=1}^{\infty}((-1)^kc_k + d_k).

    Your answer should be the formulas for c_k and d_k separated by commas.Both terms should be of the form \frac{c}{k^p} for some constants c and p. Note that c_k should be first."

    If anyone has any hints as to how to approach this (for example, is there a simple matter of algebraic finagling to be done or some other "hidden" method?), I would be grateful.

    Thanks

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  2. #2
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    Have you tried c_k  = \frac{8}{{k^{\frac{1}{2}} }}\,\& \,d_k  = \frac{{ - 7}}{k}?
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  3. #3
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    Quote Originally Posted by Plato View Post
    Have you tried c_k  = \frac{8}{{k^{\frac{1}{2}} }}\,\& \,d_k  = \frac{{ - 7}}{k}?

    Oh, no. Even though those seem obvious, I hadn't tried any particular p-series, because I wasn't sure what the problem was asking. But, it appears those are the correct answers.
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by auslmar View Post
    Oh, no. Even though those seem obvious, I hadn't tried any particular p-series, because I wasn't sure what the problem was asking. But, it appears those are the correct answers.
    how did Plato see there solutions?

    \sum_{k=1}^{\infty}\frac{8(-1)^kk-7(k)^\frac{1}{2}}{k^\frac{3}{2}} = \sum_{k = 1}^\infty \bigg( \frac {8(-1)^kk}{k^{\frac 32}} - \frac {7k^{\frac 12}}{k^{\frac 32}} \bigg) = \sum_{k = 1}^\infty \bigg( (-1)^k {\color{red}\frac {8k}{k^{\frac 32}}} - {\color{red}\frac {7k^{\frac 12}}{k^{\frac 32}}} \bigg).

    now do you see where they came from?
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  5. #5
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    Quote Originally Posted by Jhevon View Post
    how did Plato see there solutions?

    \sum_{k=1}^{\infty}\frac{8(-1)^kk-7(k)^\frac{1}{2}}{k^\frac{3}{2}} = \sum_{k = 1}^\infty \bigg( \frac {8(-1)^kk}{k^{\frac 32}} - \frac {7k^{\frac 12}}{k^{\frac 32}} \bigg) = \sum_{k = 1}^\infty \bigg( (-1)^k {\color{red}\frac {8k}{k^{\frac 32}}} - {\color{red}\frac {7k^{\frac 12}}{k^{\frac 32}}} \bigg).

    now do you see where they came from?
    Yeah, I can see now. It just wasn't occurring to me before.
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  6. #6
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    Quote Originally Posted by auslmar View Post
    those seem obvious, I hadn't tried any particular p-series, because I wasn't sure what the problem was asking. But, it appears those are the correct answers.
    I have some real concerns about the series of problems you have posted.
    It appears that whatever computer program is validating your answers has some real limitations. Therefore, I must say that I not sure what you are learning from this exercise. I hope that I am wrong.
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  7. #7
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    Quote Originally Posted by Plato View Post
    I have some real concerns about the series of problems you have posted.
    It appears that whatever computer program is validating your answers has some real limitations. Therefore, I must say that I not sure what you are learning from this exercise. I hope that I am wrong.
    Yes, I agree that the assignments and the system aren't designed as great exercises.
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