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Infinite sum converge to what value?

  1. Apr 9, 2007 #1
    1. The problem statement, all variables and given/known data
    The infinite series (-1)^n(x/n) from n=1 converges. But what is the specific value of it?
     
  2. jcsd
  3. Apr 9, 2007 #2

    Gib Z

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    [tex]\ln(1+x) = \sum^{\infty}_{n=0} \frac{(-1)^n}{n+1} x^{n+1}[/tex]
    [tex]\sum_{n=1}^{\infty} (-1)^n\frac{x}{n} = x\sum_{n=1}^{\infty} \frac{(-1)^n}{n}=x\log_e 2[/tex]
     
  4. Apr 9, 2007 #3
    You have put x=1 so it should just be ln(2)?
     
  5. Apr 9, 2007 #4

    Gib Z

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    Yup. Exactly.
     
  6. Apr 9, 2007 #5
    But ln(2)>0 and [tex]\sum_{n=1}^{\infty} (-1)^n\frac{x}{n}<0[/tex] since the first term is negative and has the largest magnitude so will dominate the series. The series should equal -ln(2) so you may have made an error with your series manipulation.
     
    Last edited: Apr 9, 2007
  7. Apr 10, 2007 #6

    Gib Z

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    Yea sorry about that >.< I made a mistake with the starts of the series, some were n=1 and others n=0, and I didn't handle them well. But youve got the idea
     
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