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Math Help - Power Series

  1. #1
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    Power Series

    I need to find the interval on convergence of the given power series

    I am trying to do summation of nx^x
    I am using s(n+1)/s(n) to solve the problem, but am having trouble getting an answer. Thanks for the help.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by davecs77 View Post
    I need to find the interval on convergence of the given power series

    I am trying to do summation of nx^x
    I am using s(n+1)/s(n) to solve the problem, but am having trouble getting an answer. Thanks for the help.
    We wish to find the radius of convergence for \sum_{n = 1}^{ \infty} nx^n

    We will proceed by the Ratio test (since that's what you wanted, I'd probably do the root test here).

    Let s_n = nx^n

    by the Ratio test, we have convergence if \lim_{n \to \infty} \left|  \frac {s_{n + 1}}{s_n}\right| < 1

    Now, \lim_{n \to \infty} \left| \frac {(n + 1)x^{n + 1}}{nx^n} \right| = \lim_{n \to \infty} \left| \frac {n + 1}{n} \cdot x \right| = |x|

    Thus we have convergence if |x| < 1

    so our radius of convergence is 1
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  3. #3
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    Quote Originally Posted by Jhevon View Post
    i believe it should be x^n, am i right?

    oops I messed up..actually it should be
    summation of nx^(n)
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by davecs77 View Post
    oops I messed up..actually it should be
    summation of nx^(n)
    yes, i realize that, i already gave you the required solution
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