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Homework Help: Radius of convergence

  1. Dec 12, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the radius of convergence of [tex]\sum[/tex] n=1 to ∞ of [(-1)^(n-1) x^(n)/(n)] and give a formula for the value of the series at the right hand endpoint.

    2. Relevant equations


    3. The attempt at a solution

    Not really sure how to start this. I know I'm supposed to use the root test or the ratio test and then probably Abel's Thm for series. but the n-1 in the series is throwing me off. I've never seen that before. Any help is appreciated.
     
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  3. Dec 12, 2009 #2

    Dick

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    Your series is just f(x)=x-x^2/2+x^3/3+... Just use the ratio test to find the radius of convergence. To find the actual value at the endpoint you could just ask yourself is you have seen that kind of series before. If it doesn't look familiar, try finding f'(x) and see if that series looks familiar.
     
  4. Dec 13, 2009 #3
    So I took f'(x) = 1 - x + x^2 - x^3 + ..... so, the endpoint is going to look like x^(n+1)? I need a formula for the value at the right hand endpoint..would that work?
     
  5. Dec 13, 2009 #4

    Dick

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    No, you have to recognize the series, my point to looking at f'(x) was that it's geometric, it's 1/(1+x). if that's f'(x), what's f(x)?
     
  6. Dec 14, 2009 #5
    If f'(x) = 1/(1+x) then f(x) = ln(x+1)?
     
  7. Dec 14, 2009 #6

    Dick

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    ln(x+1)+C, you mean. You have to fix the constant C by finding f(0). But yes, that's the idea.
     
  8. Dec 15, 2009 #7
    Alright, thank you.
     
  9. Dec 16, 2009 #8

    HallsofIvy

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    The ratio test applies to a positive series and a power series converges absolutely inside its "radius of convergence".

    Take the absolute value (drop the "[itex](-1)^{n-1}[/itex]") and then use the ratio test.
     
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