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Homework Help: Series problem help

  1. Oct 6, 2006 #1
    I want to evaluate [tex] \int \frac{\sin x}{x} [/tex].

    So [tex] \sin x = \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{2n+1}}{(2n+1)!} [/tex]. Therefore [tex] \frac{\sin x}{x} = \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{2n}}{(2n+1)!} [/tex]. So would that mean:

    [tex] \int \frac{\sin x}{x} = C + \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{2n+1}}{2n+1(2n+1)!} [/tex] would be absolutely convergent (i.e. [tex] R = \infty [/tex])?

    Thanks
     
  2. jcsd
  3. Oct 6, 2006 #2

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    How do you know that series is absolutely convergent?
     
  4. Oct 6, 2006 #3
    I would use the ratio test [tex] |\frac{ a_{n+1}}{a_{n}}| [/tex]. If the limit as [tex] n\rightarrow \infty [/tex] is less than 1, then the series is absolutely convergent. Ok so I guess it is then.
     
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