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Math Help - Power Series Convergence

  1. #1
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    Power Series Convergence

    Hello!
    I want to find the radius of convergence of the following power series:

    \sum^{\infty}_{n=1} \frac{1}{\sqrt{n}} x^n

    using the ratio test, \frac{S_{n+1}}{S_{n}}:

    \frac{1}{\sqrt{n+1}} x^{n+1} / \frac{1}{\sqrt{n}} x^n

    = \sqrt{\frac{n}{n+1}} |x|

    Now I'm confused, is the limit as n->infinity = 1 or 0?

    Because lim_{n \rightarrow \infty} \sqrt{\frac{n}{n+1}}=0
    0.|x| = 0 !?

    Or do we just ignore the \frac{1}{\sqrt{n}} part and say the interval of convergence is |x|<1 and the radius is 1 ?
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  2. #2
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    Quote Originally Posted by Roam View Post
    Hello!
    I want to find the radius of convergence of the following power series:

    \sum^{\infty}_{n=1} \frac{1}{\sqrt{n}} x^n

    using the ratio test, \frac{S_{n+1}}{S_{n}}:

    \frac{1}{\sqrt{n+1}} x^{n+1} / \frac{1}{\sqrt{n}} x^n

    = \sqrt{\frac{n}{n+1}} |x|

    Now I'm confused, is the limit as n->infinity = 1 or 0?

    Because lim_{n \rightarrow \infty} \sqrt{\frac{n}{n+1}}=0
    0.|x| = 0 !?

    Or do we just ignore the \frac{1}{\sqrt{n}} part and say the interval of convergence is |x|<1 and the radius is 1 ?
    \lim_{n \to \infty}\sqrt{\frac{n}{n+1}}


    \lim_{n \to \infty}\sqrt{\frac{1}{1+\frac{1}{n}}}=\sqrt{\frac{  1}{1+0}}=1
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