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Math Help - calculating a power series

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    calculating a power series

    calculate a power series for ln (1+x^4), it's radius of convergence, and its interval of convergence

    i'm really unsure of where to start/how to do this one, and i'm quite overwhelmed with these problemsplease help- i wouldn't be asking so much, but this was assigned last minute and i really need it!
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  2. #2
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    Quote Originally Posted by buttonbear View Post
    calculate a power series for ln (1+x^4), it's radius of convergence, and its interval of convergence

    i'm really unsure of where to start/how to do this one, and i'm quite overwhelmed with these problemsplease help- i wouldn't be asking so much, but this was assigned last minute and i really need it!
    you should already know the power series for \ln(1+x) ...

    \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ...

    \ln(1+x^3) = x^3  - \frac{x^6}{2} + \frac{x^9}{3} - \frac{x^{12}}{4} + ...

    determine the nth term and use the ratio test to find the radius and interval of convergence.
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    well, i wish i could say i learned it, but we're kind of left to struggle on our own...

    so, for ln (1+x^4), would it be = x^4  - \frac{x^8}{2} + \frac{x^12}{3} - \frac{x^{16}}{4} + ...?
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  4. #4
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    Quote Originally Posted by buttonbear View Post
    well, i wish i could say i learned it, but we're kind of left to struggle on our own...

    so, for ln (1+x^4), would it be = x^4  - \frac{x^8}{2} + \frac{x^12}{3} - \frac{x^{16}}{4} + ...?
    ok ... what's the nth term?

    use the ratio test to find the interval of convergence ...

    \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1
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    the nth term is...  \frac{x^{4n}}{n}?

    edit: i mean..  (-1^{n-1})\frac{x^{4n}}{n}

    editx2: actually..i'm not sure what the -1 is raised to, now that i think of it..
    Last edited by buttonbear; April 27th 2009 at 10:24 PM.
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