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Math Help - Power Series

  1. #1
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    Power Series

    x* \sum_{1 }^{ \infty  } na_nx^{n-1} +\sum_{0 }^{ \infty  }a_nx^n

    I have to rewrite this expression with one just involving x^n apparently the answer is \sum_{0 }^{ \infty  }(n+1)a_nx^n but i dont know how to combine the two series.. any help?
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  2. #2
    Member Mentia's Avatar
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    So if the limits of your sum are the same and are over the same parameter then you can just put them together:

    Since you are not summing over x you can just put it inside:
    <br />
x* \sum_{1 }^{ \infty } na_nx^{n-1} +\sum_{0 }^{ \infty }a_nx^n =  \sum_{1 }^{ \infty } na_nx^{n} +\sum_{0 }^{ \infty }a_nx^n<br />

    Then you can change the limits of the first sum if you subtract the zero term afterwards:

    \sum_{1 }^{ \infty } na_nx^{n} +\sum_{0 }^{ \infty }a_nx^n = \sum_{0 }^{ \infty } na_nx^{n} - 0a_0x^{0}+\sum_{0 }^{ \infty }a_nx^n = \sum_{0 }^{ \infty } na_nx^{n} +\sum_{0 }^{ \infty }a_nx^n<br />

    Now we have the same limits over the same parameter so we can just put them together:

    \sum_{0 }^{ \infty } na_nx^{n} +\sum_{0 }^{ \infty }a_nx^n = \sum_{0 }^{ \infty } na_nx^{n} + a_nx^n = \sum_{0 }^{ \infty } a_nx^{n}(n+1)
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