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Math Help - Power Series

  1. #1
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    Power Series

    a function is defined by

    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>=0

    Find the interval of convergence of the series and find an explicit formula for f(x)
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by amiv4 View Post
    a function is defined by

    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>=0

    Find the interval of convergence of the series and find an explicit formula for f(x)
    We have


    \begin{aligned}1+2x+x^2+2x^3+\cdots&=\sum_{n=0}^{\  infty}x^{2n}+2\sum_{n=0}^{\infty}x^{2n+1}\\<br />
&=\sum_{n=0}^{\infty}\bigg[x^{2n}+2x^{2n+1}\bigg]\\<br />
&=\sum_{n=0}^{\infty}(2x+1)x^{2n}\end{aligned}

    So applying the root test we get

    \begin{aligned}\lim_{n\to\infty}\sqrt[n]{(2x+1)x^{2n}}&=x^2\lim_{n\to\infty}\sqrt[n]{2x+1}\\<br />
&=x^2<1\\<br />
&\implies{|x|<1}\end{aligned}


    I think its pretty clear that this diverges at both endpoints, so we may conclude that the interval of convergence is (-1,1). And if you are interested \forall{x}\in(-1,1)\sum_{n=0}^{\infty}(2x+1)x^{2n}=\frac{2x+1}{1-x^2}
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  3. #3
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    what other tests could i use besides the root test
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by amiv4 View Post
    what other tests could i use besides the root test
    Ratio test would also work. You could also note that \sum_{n=0}^{\infty}x^{2n}+2\sum_{n=0}^{\infty}x^{2  n+1} are the sum of two geometric series, and that each is only convergent on (-1,1), so the summation of them combined's IOC would be (-1,1)\cup(-1,1)=(-1,1)
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