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Math Help - Power series limit help

  1. #1
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    Power series limit help

    Hi im having trouble understanding part of the power series. Im trying to find the radius of convergence using the ratio test, the initial problem is the sum of k=1 as k goes to infinity of x^k/(1+k^2). When i use the ratio test i get it down to xk^2/(k+1)^2, now my book goes from this to just the absolute value of x. If you expand (k+1)^2 the k^2's cancel but what happens to the 2k+1 are you allowed to just drop it or am i missing something?
    Thanks for the help.
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    Quote Originally Posted by cowboys111 View Post
    Hi im having trouble understanding part of the power series. Im trying to find the radius of convergence using the ratio test, the initial problem is the sum of k=1 as k goes to infinity of x^k/(1+k^2). When i use the ratio test i get it down to xk^2/(k+1)^2, now my book goes from this to just the absolute value of x. If you expand (k+1)^2 the k^2's cancel but what happens to the 2k+1 are you allowed to just drop it or am i missing something?
    I fear that you may be missing a lot.
    \frac{{k^2 }}{{\left( {1 + k} \right)^2 }} = \left( {\frac{1}{{\frac{1}{k} + 1}}} \right)^2  \to 1
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  3. #3
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    Quote Originally Posted by cowboys111 View Post
    Hi im having trouble understanding part of the power series. Im trying to find the radius of convergence using the ratio test, the initial problem is the sum of k=1 as k goes to infinity of x^k/(1+k^2). When i use the ratio test i get it down to xk^2/(k+1)^2, now my book goes from this to just the absolute value of x. If you expand (k+1)^2 the k^2's cancel but what happens to the 2k+1 are you allowed to just drop it or am i missing something?
    Thanks for the help.
    \sum_{k=1}^{\infty}\frac{x^k}{1+k^2}

    \lim_{k \to \infty} \left| \frac{\frac{x^{k+1}}{1+(k+1)^2}}{\frac{x^k}{1+k^2}  } \right|= \lim_{k \to \infty} \left| \frac{x^{k+1}}{k^2+2k+2} \cdot \frac{k^2+1}{x^k} \right|=\lim_{k \to \infty} \left| \frac{x(1+\frac{1}{k^2})}{1+\frac{2}{k}+\frac{2}{k  ^2}} \right|=|x|

    the 2nd to last step comes from multiplying the numerator and denominator by \frac{1}{k^2}
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