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Math Help - Representing functions as Power Series

  1. #1
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    Representing functions as Power Series

    Hi, can someone please help me with this problem? I started it but couldn't finish. Thanks a lot!!

    f(x) = 3x/1-5x^3
    = 3x * 1/1-5x^3
    = 3x sigma (5x^3)^n...
    Now I think I need to integrate both sides. but I am not sure if what I have done is correct so far?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by jsu03 View Post
    Hi, can someone please help me with this problem? I started it but couldn't finish. Thanks a lot!!

    f(x) = 3x/1-5x^3
    = 3x * 1/1-5x^3
    = 3x sigma (5x^3)^n...
    Now I think I need to integrate both sides. but I am not sure if what I have done is correct so far?
    The singular 3x in the numerator is meaningless(why?). So let us consider \frac{1}{1-5x^3} well if we rewrite this as \frac{1}{1-z}\quad z=5x^3 then it is clear from the geometric sum \frac{1}{1-u}=\sum_{n=0}^{\infty}u^n that \frac{1}{1-5x^3}=\sum_{n=0}^{\infty}\left(5x^3\right)^n=\sum_  {n=0}^{\infty}5^nx^{3n}. Which is precisely what you got. So whats the problem?
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  3. #3
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    Now to finish the problem I need to integrate both sides
    Here is what I did:

    integral 1/1- 5x^3 = integral 5^n x^3n
    sigma 5x^3n + sigma 5^n x^5n+1/5^n+1
    = sigma 5^n x^5n+1/ 5n+1 + C

    so radius of convergence = 5
    interval of convergence = (-5,5)

    So my question is are these calculations correct cause I am still confuse. Thanks again.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by jsu03 View Post
    Now to finish the problem I need to integrate both sides
    Here is what I did:

    integral 1/1- 5x^3 = integral 5^n x^3n
    sigma 5x^3n + sigma 5^n x^5n+1/5^n+1
    = sigma 5^n x^5n+1/ 5n+1 + C

    so radius of convergence = 5
    interval of convergence = (-5,5)

    So my question is are these calculations correct cause I am still confuse. Thanks again.
    I am completely lost. The power series representation of f(x)=\frac{3x}{1-5x^3} is 3x\cdot\sum_{n=0}^{\infty}5^nx^{3n}...why are you integrating?!?!
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