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Math Help - Finding the formula

  1. #1
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    Finding the formula

    Hi, this question involves finding an explicit formula for the given f(x):

    f(x)=1+2x+x^2+2x^3+x^4+.... that is, its coefficients are c_{2n=1} and c_{2n+1}=2 for all n\geq0

    However, I only knw how to work from the formla to get the power series but not otherwise. Please help.

    The answer is f(x)=\frac{1+2x}{1-x^2}
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  2. #2
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    Quote Originally Posted by noob mathematician View Post
    Hi, this question involves finding an explicit formula for the given f(x):

    f(x)=1+2x+x^2+2x^3+x^4+.... that is, its coefficients are c_2n=1 and c_2n+1=2 for all n\leq0

    However, I only knw how to work from the formla to get the power series but not otherwise. Please help.

    The answer is f(x)=\frac{1+2x}{1-x^2}
    Re-write your series as (i) the even terms and (ii) the odd terms with a factor of 2x taken out

     <br />
1 + x^2 + x^4 + x^6 + \cdots + 2x\left(1 + x^2 + x^4 + x^6 + \cdots \right)<br />

    Then use the geometric power series

     <br />
\frac{1}{1-x} = 1 + x + x^2 + x^3 \cdots.<br />
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  3. #3
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    Hello,

    Can't you just split the sum ?

    f(x)=\sum_{k=0}^\infty x^{2k}+2\sum_{k=0}^\infty x^{2k+1}=\sum_{k=0}^\infty x^{2k}+2x\sum_{k=0}^\infty x^{2k}=(1+2x)\sum_{k=0}^\infty x^{2k}
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