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Math Help - Determine the Interval of Convergence

  1. #1
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    Determine the Interval of Convergence

    Determine the interval of convergence

    \sum \frac{-1^k(x+\frac{\pi}{2})^{2k+1}}{2k+1}

    I used the ratio test and got it down to

    \frac{1}{2}\lim= -(x+\frac{\pi}{2})^3

    but I think I messed up somewhere or maybe I shouldn't use the ratio test and maybe try something different.
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  2. #2
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    Is this your series: \sum_{k=0}^{\infty} \frac{(-1)^k (x+\frac{\pi}{2})^{2k+1}}{2k+1}

    Using the ratio test:
    \begin{aligned} \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| & = \lim_{k \to \infty} \left|\frac{ (-1)^{k+1} (x+\frac{\pi}{2})^{2(k+1)+1} }{2(k+1)+1} \ \cdot \ \frac{2k+1}{(-1)^k (x + \frac{\pi}{2})^{2k+1} } \right| \\ & = \lim_{k \to \infty} \left| \frac{ (x+\frac{\pi}{2})^{2k+3} }{2k+3} \ \cdot \ \frac{2k+1}{(x+ \frac{\pi}{2})^{2k+1} }\right| \end{aligned}
    \begin{aligned} {\color{white}\lim_{k \to \infty} \left| \frac{a_{n+1}}{a_n} \right|} & = \lim_{k \to \infty} \left| (x+\frac{\pi}{2})^2 \cdot \frac{2k+1}{2k+3} \right| \\ & = \left(x+ \frac{\pi}{2}\right)^2 \cdot \underbrace{\lim_{k \to \infty} \left|\frac{2k+1}{2k+3}\right|}_{=1} \end{aligned}

    So by the ratio test, your series absolutely converges and thus converges if \lim_{k \to \infty} \frac{a_{k+1}}{a_k} = \left(x + \frac{\pi}{2}\right)^2 {\color{red}<} \ 1

    Simply solve.
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by kl.twilleger View Post
    Determine the interval of convergence

    \sum \frac{-1^k(x+\frac{\pi}{2})^{2k+1}}{2k+1}{\color{red}=\su  m{a_k}}

    I used the ratio test and got it down to

    \frac{1}{2}\lim= -(x+\frac{\pi}{2})^3

    but I think I messed up somewhere or maybe I shouldn't use the ratio test and maybe try something different.
    You don't have to use the Ratio test. Just note that a_k\sim\left[\left(x+\frac{\pi}{2}\right)^2\right]^k

    Which is a geometric series and converges iff the value raised to the n is less then one.

    P.S. Dont forget to check the endpoints of your IOC

    Also note that for every x that makes this seris converge it is euqal to \arctan\left(x+\frac{\pi}{2}\right)
    Last edited by Mathstud28; November 13th 2008 at 02:10 PM. Reason: typo
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  4. #4
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    Quote Originally Posted by Mathstud28 View Post
    You don't have to use the Ratio test. Just note that a_k\sim\left[\left(x+\frac{\pi}{2}\right)^2\right]
    What test is "needed" I think really depends on the level of rigorousness expected for the problem. Your observation about a_k is a good intuitive insight but without justification it's useless in the context of a proof. I'm sure you can easily show why what you said is true but like I said it all depends on the rigorousness required of a solution.
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Jameson View Post
    What test is "needed" I think really depends on the level of rigorousness expected for the problem. Your observation about a_k is a good intuitive insight but without justification it's useless in the context of a proof. I'm sure you can easily show why what you said is true but like I said it all depends on the rigorousness required of a solution.
    Yes, I understand what you are saying. A lot of the things I say similar to this are not meant to be taken as solutions, or even as what I would put down. They are more a way to rationalize why the answer is what it is. It also enables the student to form a more intuitive insight into mathematics as a whole. So yes you are correct, this should never be given as a solution especially in a formal situation like a quiz or test but I beleive that making these "intuitive insights" is the key to advanced mathematical success.
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